Neural-PDE-Solver

PDE: Partial Differentiable Equation

Neural Operators: Learning nonlinear mappings between function spaces.

Contributed by Chunyang Zhang.

Content

1. Survey
2. Model
2.1 PINN	2.2 DeepONet
2.3 Fourier Operator	2.4 Graph Network
2.5 Green Function	2.6 Finite Element
2.7 Convolution	2.8 AutoEncoder
2.9 Neural Operator	2.10 Machine Learning
2.11 Identification	2.12 Inverse Design
2.13 Neural ODE	2.14 Large Model
3. Mechanism
3.1 Library	3.2 Analysis
3.3 Domain Adaptation	3.4 Loss Function
3.5 Sampling	3.6 Mesh
3.7 Decomposition	3.8 Disentangle
3.9 Solver	3.10 AutoML
3.11 Neural Implicit Flow	3.12 Uncertainty Quantification
3.13 Generative Model	3.14 Transformer
3.15 Theory	3.16 Gaussian Process
3.17 Variation	3.18 Bayesian
3.19 Latent Space	3.20 Lagrangian
3.21 Multi Scale	3.22 Multi Fidelity
3.23 Multi Grid	3.24 Active Learning
4. Applications
4.1 Optimization	4.2 Fluid
4.3 Cybernetics	4.4 Climate
4.5 Mechanics	4.6 Robotics
4.7 Physics	4.8 Image
4.9 Molecules	4.10 Reconstruction
4.11 Quantum	4.12 Game Theory
4.13 Industry	4.14 Economics

Survey Papers

1. Physics-informed machine learning. Nature Reviews Physics, 2021. paper

   George Em Karniadakis, Ioannis G. Kevrekidis, Lu Lu, Paris Perdikaris, Sifan Wang, and Liu Yang.

2. Neural operator: Learning maps between function spaces. arXiv, 2021. paper

   Nikola Kovachki, Zongyi Li, Burigede Liu, Kamyar Azizzadenesheli, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar.

3. Physics-informed machine learning approach for augmenting turbulence models: A comprehensive framework. Physical Review Fluids, 2018. paper

   Jinlong Wu, Heng Xiao, and Eric Paterson.

4. Integrating scientific knowledge with machine learning for engineering and environmental systems. ACM Computing Surveys, 2023. paper

   Jared Willard, Xiaowei Jia, Shaoming Xu, Michael Steinbach, and Vipin Kumar.

5. Physical laws meet machine intelligence: Current developments and future directions. Artificial Intelligence Review, 2022. paper

   Temoor Muther, Amirmasoud Kalantari Dahaghi, Fahad Iqbal Syed, and Vuong Van Pham.

6. A comprehensive and fair comparison of two neural operators (with practical extensions) based on FAIR data. Computer Methods in Applied Mechanics and Engineering, 2022. paper

   Lu Lu, Xuhui Meng, Shengze Cai, Zhiping Mao, Somdatta Goswami, Zhongqiang Zhang, and George Em Karniadakis.

7. Scientific machine learning through physics–informed neural networks: Where we are and what’s next. Beyond Traditional AI: The Impact of Machine Learning on Scientific Computing, 2022. book

   MingyuanYang and John T.Foster

8. When physics meets machine learning: A survey of physics-informed machine learning. arXiv, 2022. paper

   Chuizheng Meng, Sungyong Seo, Defu Cao, Sam Griesemer, and Yan Liu.

9. Physics-guided, physics-informed, and physics-encoded neural networks in scientific computing. arXiv, 2022. paper

   Salah A. Faroughi, Nikhil M. Pawar, C´elio Fernandes, Subasish Das, Nima K. Kalantari, and Seyed Kourosh Mahjour.

10. Physics-informed machine learning: A survey on problems, methods and applications. arXiv, 2022. paper

    Zhongkai Hao, Songming Liu, Yichi Zhang, Chengyang Ying, Yao Feng, Hang Su, and Jun Zhu.

11. An overview on deep learning-based approximation methods for partial differential equations. arXiv, 2020. paper

    Christian Beck, Martin Hutzenthaler, Arnulf Jentzen, and Benno Kuckuck.

12. Three ways to solve partial differential equations with neural networks—A review. GAMM‐Mitteilungen, 2021. paper

    Jan Blechschmidt and Oliver G. Ernst.

13. Combining machine learning and domain decomposition methods for the solution of partial differential equations—A review. GAMM‐Mitteilungen, 2021. paper

    Alexander Heinlein, Axel Klawonn, Martin Lanser, and Janine Weber.

14. Physics-guided, physics-informed, and physics-encoded neural networks in scientific computing. arXiv, 2022. paper

    Salah A Faroughi, Nikhil Pawar, Celio Fernandes, Subasish Das, Nima K. Kalantari, and Seyed Kourosh Mahjour.

15. Partial differential equations meet deep neural networks: A survey. arXiv, 2022. paper

    Shudong Huang, Wentao Feng, Chenwei Tang, and Jiancheng Lv.

16. Solving differential equations with Deep Learning: A beginner's guide. arXiv, 2023. paper

    Luis Medrano Navarro, Luis Martín Moreno, and Sergio G Rodrigo.

17. Deep learning algorithms for solving differential equations: a survey. Journal of Experimental & Theoretical Artificial Intelligence, 2023. paper

    Harender Kumara and Neha Yadav.

18. An expert's guide to training physics-informed neural networks. arXiv, 2023. paper

    Sifan Wang, Shyam Sankaran, Hanwen Wang, and Paris Perdikaris.

19. A survey on physics informed reinforcement learning: Review and open problems. arXiv, 2023. paper

    Chayan Banerjee, Kien Nguyen, Clinton Fookes, and Maziar Raissi.

20. Neural operators for accelerating scientific simulations and design. arXiv, 2023. paper

    Kamyar Azzizadenesheli, Nikola Kovachki, Zongyi Li, Miguel Liu-Schiaffini, Jean Kossaifi, and Anima Anandkumar.

21. Machine learning and domain decomposition methods -- A survey. arXiv, 2023. paper

    Axel Klawonn, Martin Lanser, and Janine Weber.

22. The transformative potential of machine learning for experiments in fluid mechanics. Nature Reviews Physics, 2023. paper

    Ricardo Vinuesa, Steven L. Brunton, and Beverley J. McKeon.

23. Physics-informed machine learning for reliability and systems safety applications: State of the art and challenges. Reliability Engineering & System Safety, 2023. paper

    Yanwen Xu, Sara Kohtz, Jessica Boakye, Paolo Gardoni, and Pingfeng Wang.

Model

PINN

1. Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations. Science, 2020. paper

   Raissi Maziar, Alireza Yazdani, and George Em Karniadakis.

2. Deep hidden physics models: Deep learning of nonlinear partial differential equations. JMLR, 2018. paper

   Maziar Raissi.

3. A universal PINNs method for solving partial differential equations with a point source. IJCAI, 2022. paper

   Xiang Huang, Hongsheng Liu, Beiji Shi, Zidong Wang, Kang Yang, Yang Li, Min Wang, Haotian Chu, Jing Zhou, Fan Yu, Bei Hua, Bin Dong, and Lei Chen.

4. Parallel physics-informed neural networks via domain decomposition. JCP, 2021. paper

   Khemraj Shukla, Ameya D.Jagtap, and George Em Karniadakis.

5. Kolmogorov n–width and Lagrangian physics-informed neural networks: A causality-conforming manifold for convection-dominated PDEs. Computer Methods in Applied Mechanics and Engineering, 2023. paper

   Rambod Mojgani, Maciej Balajewicz, and Pedram Hassanzadeh.

6. Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks. Computer Methods in Applied Mechanics and Engineering, 2022. paper

   N.Sukumar and Ankit Srivastava.

7. Physics-informed multi-LSTM networks for meta-modeling of nonlinear structures. Computer Methods in Applied Mechanics and Engineering, 2020. paper

   Ruiyang Zhang, Yang Liu, and Hao Sun.

8. Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems. Computer Methods in Applied Mechanics and Engineering, 2022. paper

   Jeremy Yu, Lu Lu, Xuhui Meng, and George Em Karniadakis.

9. Multi-output physics-informed neural networks for forward and inverse PDE problems with uncertainties. Computer Methods in Applied Mechanics and Engineering, 2022. paper

   MingyuanYang and John T.Foster

10. PPINN: Parareal physics-informed neural network for time-dependent PDEs. Computer Methods in Applied Mechanics and Engineering, 2020. paper

    Xuhui Meng, Zhen Li, Dongkun Zhang, and George Em Karniadakis.

11. CAN-PINN: A fast physics-informed neural network based on coupled-automatic–numerical differentiation method. Computer Methods in Applied Mechanics and Engineering, 2022. paper

    Pao-Hsiung Chiu, Jian Cheng Wong, Chinchun Ooi, My Ha Dao, and Yew-Soon Ong.

12. Derivative-informed projected neural networks for high-dimensional parametric maps governed by PDEs. Computer Methods in Applied Mechanics and Engineering, 2022. paper

    Thomas O’Leary-Roseberry, Umberto Villa, Peng Chen, and Omar Ghattas.

13. Physics-augmented learning: A new paradigm beyond physics-informed learning. NIPS, 2021. paper

    Ziming Liu, Yuanqi Du, Yunyue Chen, and Max Tegmark.

14. Data-driven vector soliton solutions of coupled nonlinear Schrödinger equation using a deep learning algorithm. Physics Letters A, 2021. paper

    Yifan Mo, Liming Ling, and Delu Zeng.

15. Solving Benjamin–Ono equation via gradient balanced PINNs approach. The European Physical Journal Plus, 2022. paper

    Xiangyu Yang and Zhen Wang.

16. Robust learning of physics informed neural networks. arXiv, 2021. paper

    Chandrajit Bajaj, Luke McLennan, Timothy Andeen, and Avik Roy.

17. Learning physics-informed neural networks without stacked back-propagation. AISTATS, 2023. paper

    Di He, Wenlei Shi, Shanda Li, Xiaotian Gao, Jia Zhang, Jiang Bian, Liwei Wang, and Tieyan Liu.

18. NeuralPDE: Automating physics-informed neural networks (PINNs) with error approximations. arXiv, 2021. paper

    Kirill Zubov, Zoe McCarthy, Yingbo Ma, Francesco Calisto, Valerio Pagliarino, Simone Azeglio, Luca Bottero, Emmanuel Luján, Valentin Sulzer, Ashutosh Bharambe, Nand Vinchhi, Kaushik Balakrishnan, Devesh Upadhyay, and Chris Rackauckas.

19. Physics informed RNN-DCT networks for time-dependent partial differential equations. ICCS, 2022. paper

    Benjamin Wu, Oliver Hennigh, Jan Kautz, Sanjay Choudhry, and Wonmin Byeon.

20. Theory-guided physics-informed neural networks for boundary layer problems with singular perturbation. JCP, 2022. paper

    Amirhossein Arzani, Kevin W.Cassel, and Roshan M.D'Souza.

21. A-PINN: Auxiliary physics informed neural networks for forward and inverse problems of nonlinear integro-differential equations. JCP, 2022. paper

    Lei Yuan, Yiqing Ni, Xiangyun Deng, and Shuo Hao.

22. A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: Comparison with finite element method. Computer Methods in Applied Mechanics and Engineering, 2022. paper

    Shahed Rezaei, Ali Harandi, Ahmad Moeineddin, Baixiang Xua, and Stefanie Reese.

23. Physics-informed neural networks combined with polynomial interpolation to solve nonlinear partial differential equations. Computers & Mathematics with Applications, 2023. paper

    Siping Tang, Xinlong Feng, Wei Wu, and Hui Xu.

24. A novel sequential method to train physics informed neural networks for Allen Cahn and Cahn Hilliard equations. Computer Methods in Applied Mechanics and Engineering, 2022. paper

    Revanth Mattey and Susanta Ghosh.

25. RPINNs: Rectified-physics informed neural networks for solving stationary partial differential equations. Computers and Fluids, 2022. paper

    Pai Peng, Jiangong Pan, Hui Xu, and Xinlong Feng.

26. A-WPINN algorithm for the data-driven vector-soliton solutions and parameter discovery of general coupled nonlinear equations. Physica D: Nonlinear Phenomena, 2022. paper

    Shumei Qin, Min Li, Tao Xu, and Shaoqun Dong.

27. Physics-informed neural networks with adaptive localized artificial viscosity. arXiv, 2022. paper

    E.J.R. Coutinho, M. Dall'Aqua, L. McClenny, M. Zhong, U. Braga-Neto, and E. Gildin.

28. Physics-informed neural operator for learning partial differential equations. arXiv, 2021. paper

    Zongyi Li, Hongkai Zheng, Nikola Kovachki, David Jin, Haoxuan Chen, Burigede Liu, Kamyar Azizzadenesheli, and Anima Anandkumar.

29. Anisotropic, sparse and interpretable physics-informed neural networks for PDEs. arXiv, 2022. paper

    Amuthan A. Ramabathiran and Prabhu Ramachandran.

30. Fast neural network based solving of partial differential equations. arXiv, 2022. paper

    Jaroslaw Rzepecki, Daniel Bates, and Chris Doran.

31. Discontinuity computing using physics-informed neural network. arXiv, 2022. paper

    Li Liu, Shengping Liu, Hui Xie, Fansheng Xiong, Tengchao Yu, Mengjuan Xiao, Lufeng Liu, and Heng Yong.

32. Learning differentiable solvers for systems with hard constraints. arXiv, 2022. paper

    Geoffrey Négiar, Michael W. Mahoney, and Aditi S. Krishnapriyan.

33. Momentum diminishes the effect of spectral bias in physics-informed neural networks. arXiv, 2022. paper

    Ghazal Farhani, Alexander Kazachek, and Boyu Wang.

34. Δ-PINNs: Physics-informed neural networks on complex geometries. arXiv, 2022. paper

    Francisco Sahli Costabal, Simone Pezzuto, and Paris Perdikaris.

35. Replacing automatic differentiation by Sobolev Cubatures fastens physics informed neural nets and strengthens their approximation power. arXiv, 2022. paper

    Juan Esteban Suarez Cardona and Michael Hecht.

36. FO-PINNs: A first-order formulation for physics informed neural networks. arXiv, 2022. paper

    Rini J. Gladstone, Mohammad A. Nabian, and Hadi Meidani.

37. Augmented physics-informed neural networks (APINNs): A gating network-based soft domain decomposition methodology. arXiv, 2022. paper

    Zheyuan Hu, Ameya D. Jagtap, George Em Karniadakis, and Kenji Kawaguchi.

38. Physics-informed neural networks for operator equations with stochastic data. arXiv, 2022. paper

    Paul Escapil-Inchauspé and Gonzalo A. Ruz.

39. Physics-informed neural networks with unknown measurement noise. arXiv, 2022. paper

    Philipp Pilar and Niklas Wahlstrom.

40. On the compatibility between a neural network and a partial differential equation for physics-informed learning. arXiv, 2022. paper

    Kuangdai Leng and Jeyan Thiyagalingam.

41. Pre-training strategy for solving evolution equations based on physics-informed neural networks. arXiv, 2022. paper

    Jiawei Guo, Yanzhong Yao, Han Wang, and Tongxiang Gu.

42. L-HYDRA: Multi-head physics-informed neural networks. arXiv, 2023. paper

    Zongren Zou and George Em Karniadakis.

43. PINN for dynamical partial differential equations is not training deeper networks rather learning advection and time variance. arXiv, 2023. paper

    Siddharth Rout.

44. Wavelets based physics informed neural networks to solve non-linear differential equations. Scientific Reports, 2023. paper

    Ziya Uddin, Sai Ganga, Rishi Asthana, and Wubshet Ibrahim.

45. Improved training of physics-informed neural networks using energy-based priors: A study on electrical impedance tomography. ICLR, 2023. paper

    Akarsh Pokkunuru, Pedram Rooshenas, Thilo Strauss, Anuj Abhishek, and Taufiquar Khan.

46. Adaptive weighting of Bayesian physics informed neural networks for multitask and multiscale forward and inverse problems. arXiv, 2023. paper

    Sarah Perez, Suryanarayana Maddu, Ivo F. Sbalzarini, and Philippe Poncet.

47. Efficient physics-informed neural networks using hash encoding. arXiv, 2023. paper

    Xinquan Huang and Tariq Alkhalifah.

48. Ensemble learning for physics informed neural networks: A gradient boosting approach. arXiv, 2023. paper

    Zhiwei Fang, Sifan Wang, and Paris Perdikaris.

49. On the limitations of physics-informed deep learning: Illustrations using first order hyperbolic conservation law-based traffic flow models. arXiv, 2023. paper

    Archie J. Huang and Shaurya Agarwal.

50. Achieving high accuracy with PINNs via energy natural gradients. arXiv, 2023. paper

    Johannes Müller and Marius Zeinhofer.

51. Implicit stochastic gradient descent for training physics-informed neural networks. arXiv, 2023. paper

    Ye Li, Songcan Chen, and Shengjun Huang.

52. NSGA-PINN: A multi-objective optimization method for physics-informed neural network training. arXiv, 2023. paper

    Binghang Lu, Christian B. Moya, and Guang Lin.

53. Improving physics-informed neural networks with meta-learned optimization. arXiv, 2023. paper

    Alex Bihlo.

54. MetaPhysiCa: OOD robustness in physics-informed machine learning. arXiv, 2023. paper

    S Chandra Mouli, Muhammad Ashraful Alam, and Bruno Ribeiro.

55. HomPINNs: Homotopy physics-informed neural networks for solving the inverse problems of nonlinear differential equations with multiple solutions. arXiv, 2023. paper

    Haoyang Zheng, Yao Huang, Ziyang Huang, Wenrui Hao, and Guang Lin.

56. iPINNs: Incremental learning for physics-informed neural networks. arXiv, 2023. paper

    Aleksandr Dekhovich, Marcel H.F. Sluiter, David M.J. Tax, and Miguel A. Bessa.

57. Global convergence of deep Galerkin and PINNs methods for solving partial differential equations. arXiv, 2023. paper

    Francisco Eiras, Adel Bibi, Rudy Bunel, Krishnamurthy Dj Dvijotham, Philip Torr, and M. Pawan Kumar.

58. Provably correct physics-informed neural networks. arXiv, 2023. paper

    Deqing Jiang, Justin Sirignano, and Samuel N. Cohen.

59. Predictive limitations of physics-informed neural networks in vortex shedding. arXiv, 2023. paper

    Pi-Yueh Chuang and Lorena A. Barba.

60. Residual-based error bound for physics-informed neural networks. arXiv, 2023. paper

    Shuheng Liu, Xiyue Huang, and Pavlos Protopapas.

61. Automatic boundary fitting framework of boundary dependent physics-informed neural network solving partial differential equation with complex boundary conditions. Computer Methods in Applied Mechanics and Engineering, 2023. paper

    Yuchen Xie, Yu Ma, and Yahui Wang.

62. Solving a class of multi-scale elliptic PDEs by means of Fourier-based mixed physics informed neural networks. arXiv, 2023. paper

    Xi'an Li, Jinran Wu, Zhi-Qin John Xu, and You-Gan Wang.

63. Separable physics informed neural networks. arXiv, 2023. paper

    Junwoo Cho, Seungtae Nam, Hyunmo Yang, Seok-Bae Yun, Youngjoon Hong, and Eunbyung Park.

64. Achieving high accuracy with PINNs via energy natural gradient descent. ICML, 2023. paper

    Johannes Müller and Marius Zeinhofer.

65. Gradient descent finds the global optima of two-layer physics-informed neural networks. ICML, 2023. paper

    Yihang Gao, Yiqi Gu, and Michael Ng.

66. Residual-based attention and connection to information bottleneck theory in PINNs. arXiv, 2023. paper

    Sokratis J. Anagnostopoulos, Juan Diego Toscano, Nikolaos Stergiopulos, and George Em Karniadakis.

67. Auxiliary-tasks learning for physics-informed neural network-based partial differential equations solving. arXiv, 2023. paper

    Junjun Yan, Xinhai Chen, Zhichao Wang, Enqiang Zhou, and Jie Liu.

68. Tackling the curse of dimensionality with physics-informed neural networks. arXiv, 2023. paper

    Zheyuan Hu, Khemraj Shukla, George Em Karniadakis, and Kenji Kawaguchi.

69. Solving PDEs on spheres with physics-informed convolutional neural networks. arXiv, 2023. paper

    Guanhang Lei, Zhen Lei, Lei Shi, Chenyu Zeng, and Dingxuan Zhou.

70. Solving PDEs on spheres with physics-informed convolutional neural networks. arXiv, 2023. paper

    Guanhang Lei, Zhen Lei, Lei Shi, Chenyu Zeng, and Dingxuan Zhou.

71. Tensor-compressed back-propagation-free training for (physics-informed) neural networks. arXiv, 2023. paper

    Yequan Zhao, Xinling Yu, Zhixiong Chen, Ziyue Liu, Sijia Liu, and Zheng Zhang.

72. How to select physics-informed neural networks in the absence of ground truth: A pareto front-based strategy. ICML, 2023. paper

    Zhao Wei, Jian Cheng Wong, Nicholas Wei Yong Sung, Abhishek Gupta, Chin Chun Ooi, Pao-Hsiung Chiu, My Ha Dao, and Yew-Soon Ong.

73. A gradient-enhanced physics-informed neural network (gPINN) scheme for the coupled non-fickian/non-fourierian diffusion-thermoelasticity analysis: A novel gPINN structure. EAAI, 2023. paper

    Katayoun Eshkofti and Seyed Mahmoud Hosseini.

74. Learning only on boundaries: A physics-informed neural operator for solving parametric partial differential equations in complex geometries. arXiv, 2023. paper

    Zhiwei Fang, Sifan Wang, and Paris Perdikaris.

75. Exact and soft boundary conditions in physics-informed neural networks for the variable coefficient poisson equation. arXiv, 2023. paper

    Sebastian Barschkis.

76. Investigating the ability of PINNs to solve burgers’ PDE near finite-time blowup. arXiv, 2023. paper

    Dibyakanti Kumar and Anirbit Mukherjee.

77. Correcting model misspecification in physics-informed neural networks (PINNs). arXiv, 2023. paper

    Zongren Zou, Xuhui Meng, and George Em Karniadakis.

78. On residual minimization for PDEs: Failure of PINN, modified equation, and implicit bias. arXiv, 2023. paper

    Tao Luo and Qixuan Zhou.

79. Operator learning enhanced physics-informed neural networks for solving partial differential equations characterized by sharp solutions. arXiv, 2023. paper

    Bin Lin, Zhiping Mao, Zhicheng Wang, and George Em Karniadakis.

80. PINNs-TF2: Fast and user-friendly physics-informed neural networks in TensorFlow V2. NIPS, 2023. paper

    Reza Akbarian Bafghi and Maziar Raissi.

81. Filtered partial differential equations: A robust surrogate constraint in physics-informed deep learning framework. arXiv, 2023. paper

    Dashan Zhang, Yuntian Chen, and Shiyi Chen.

82. Enhanced physics-informed neural networks with domain scaling and residual correction methods for multi-frequency elliptic problems. arXiv, 2023. paper

    Deok-Kyu Jang, Hyea Hyun Kim, and Kyungsoo Kim.

83. Physics-informed neural networks for transformed geometries and manifolds. arXiv, 2023. paper

    Samuel Burbulla.

    Zheyuan Hu, Zhouhao Yang, Yezhen Wang, George Em Karniadakis, and Kenji Kawaguchi.

84. Neuro-PINN: A hybrid framework for efficient nonlinear projection equation solutions. The International Journal for Numerical Methods in Engineering, 2023. paper

    Dawen Wu and Abdel Lisser.

85. Exactly conservative physics-informed neural networks and deep operator networks for dynamical systems. arXiv, 2023. paper

    Elsa Cardoso-Bihlo and Alex Bihlo.

86. Semi-analytic PINN methods for boundary layer problems in a rectangular domain. arXiv, 2023. paper

    Gungmin Gie, Youngjoon Hong, Chang-Yeol Jung, and Tselmuun Munkhjin.

87. PICL: Physics informed contrastive learning for partial differential equations. arXiv, 2024. paper

    Cooper Lorsung and Amir Barati Farimani.

88. Fourier warm start for physics-informed neural networks. EAAI, 2024. paper

    Ge Jin, Jian Cheng Wong, Abhishek Gupta, Shipeng Li, and Yew-Soon Ong.

89. Preconditioning for physics-informed neural networks. ICML, 2024. paper

    Songming Liu, Chang Su, Jiachen Yao, Zhongkai Hao, Hang Su, Youjia Wu, and Jun Zhu.

DeepONet

1. Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators. NMI, 2021. paper

   Lu Lu, Pengzhan Jin, Guofei Pang, Zhongqiang Zhang, and George Em Karniadakis.

2. Learning the solution operator of parametric partial differential equations with physics-informed DeepONets. SA, 2021. paper

   Wang Sifan, Hanwen Wang, and Paris Perdikaris.

3. Deep transfer operator learning for partial differential equations under conditional shift. NMI, 2022. paper

   Somdatta Goswami, Katiana Kontolati, Michael D. Shields, and George Em Karniadakis.

4. Variable-input deep operator networks. arXiv, 2022. paper

   Michael Prasthofer, Tim De Ryck, and Siddhartha Mishra.

5. MIONet: Learning multiple-input operators via tensor product. arXiv, 2022. paper

   Jeremy Yu, Lu Lu, Xuhui Meng, and George Em Karniadakis.

6. Long-time integration of parametric evolution equations with physics-informed DeepONets. arXiv, 2021. paper

   Sifan Wang and Paris Perdikaris.

7. Improved architectures and training algorithms for deep operator networks. Journal of Scientific Computing, 2022. paper

   Sifan Wang, Hanwen Wang, and Paris Perdikaris.

8. SVD perspectives for augmenting DeepONet flexibility and interpretability. arXiv, 2022. paper

   Simone Venturi and Tiernan Casey.

9. Accelerated replica exchange stochastic gradient Langevin diffusion enhanced Bayesian DeepONet for solving noisy parametric PDEs. arXiv, 2021. paper

   Guang Lin, Christian Moya, and Zecheng Zhang.

10. Bi-fidelity modeling of uncertain and partially unknown systems using DeepONet. arXiv, 2022. paper

    Subhayan De, Matthew Reynolds, Malik Hassanaly, Ryan N. King, and Alireza Doostan.

11. MultiAuto-DeepONet: A multi-resolution autoencoder DeepONet for nonlinear dimension reduction, uncertainty quantification and operator learning of forward and inverse stochastic problems. arXiv, 2022. paper

    Jiahao Zhang, Shiqi Zhang, and Guang Lin.

12. Transfer learning enhanced DeepONet for long-time prediction of evolution equations. AAAI, 2023. paper

    Wuzhe Xu, Yulong Lu, and Li Wang.

13. B-DeepONet: An enhanced Bayesian DeepONet for solving noisy parametric PDEs using accelerated replica exchange SGLD. JCP, 2023. paper

    Guang Lin, Christian Moy, and Zecheng Zhang.

14. VB-DeepONet: A Bayesian operator learning framework for uncertainty quantification. EAAI, 2023. paper

    Shailesh Garg and Souvik Chakraborty.

15. Sequential deep learning operator network (S-DeepONet) for time-dependent loads. arXiv, 2023. paper

    Jaewan Park, Shashank Kushwaha, Junyan He, Seid Koric, Diab Abueidda, and Iwona Jasiuk.

16. Asymptotic-preserving convolutional DeepONets capture the diffusive behavior of the multiscale linear transport equations. arXiv, 2023. paper

    Keke Wu, Xiong-bin Yan, Shi Jin, and Zheng Ma.

17. A hybrid decoder-DeepONet operator regression framework for unaligned observation data. arXiv, 2023. paper

    Bo Chen, Chenyu Wang, Weipeng Li, and Haiyang Fu.

18. Improving physics-informed DeepONets with hard constraints. arXiv, 2023. paper

    Rüdiger Brecht, Dmytro R. Popovych, Alex Bihlo, and Roman O. Popovych.

19. Capturing the diffusive behavior of the multiscale linear transport equations by asymptotic-preserving convolutional DeepONets. Computer Methods in Applied Mechanics and Engineering, 2023. paper

    Keke Wu, Xiong-Bin Yan, Shi Jin, and Zheng Ma.

20. DON-LSTM: Multi-resolution learning with DeepONets and long short-term memory neural networks. arXiv, 2023. paper

    Katarzyna Michałowska, Somdatta Goswami, George Em Karniadakis, and Signe Riemer-Sørensen.

21. DeepOnet based preconditioning strategies for solving parametric linear systems of equations. arXiv, 2024. paper

    Alena Kopaničáková andGeorge Em Karniadakis.

Fourier Operator

1. Fourier neural operator for parametric partial differential equations. ICLR, 2021. paper

   Zongyi Li, Nikola Borislavov Kovachki, Kamyar Azizzadenesheli, Burigede liu, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar.

2. On universal approximation and error bounds for Fourier neural operators. JMLR, 2021. paper

   Nikola Kovachki, Samuel Lanthaler, and Siddhartha Mishra.

3. HyperFNO: Improving the generalization behavior of Fourier neural operators. NIPS, 2022. paper

   Francesco Alesiani, Makoto Takamoto, and Mathias Niepert.

4. Neural operator: Graph kernel network for partial Differential equations. arXiv, 2020. paper

   Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar.

5. Fourier neural operator with learned deformations for PDEs on general geometries. arXiv, 2022. paper

   Zongyi Li, Daniel Zhengyu Huang, Burigede Liu, and Anima Anandkumar.

6. Multipole graph neural operator for parametric partial differential equations. NIPS, 2020. paper

   Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar.

7. Fast sampling of diffusion models via operator learning. ICML, 2023. paper

   Hongkai Zheng, Weili Nie, Arash Vahdat, Kamyar Azizzadenesheli, and Anima Anandkumar.

8. Factorized Fourier neural operators. ICLR, 2023. paper

   Alasdair Tran, Alexander Mathews, Lexing Xie, and Cheng Soon Ong.

9. Model inversion for spatio-temporal processes using the Fourier neural operator. NIPS, 2023. paper

   Dan MacKinlay, Dan Pagendam, Petra M. Kuhnert, Tao Cui, David Robertson, and Sreekanth Janardhanan.

10. Learning deep implicit Fourier neural operators (IFNOs) with applications to heterogeneous material modeling. Computer Methods in Applied Mechanics and Engineering, 2022. paper

    Huaiqian You, Quinn Zhang, Colton J. Ross, Chung-Hao Lee, and Yue Yu.

11. On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks. Computer Methods in Applied Mechanics and Engineering, 2021. paper

    Sifan Wang, Hanwen Wang, and Paris Perdikaris.

12. Semi-supervised learning of partial differential operators and dynamical flows. arXiv, 2022. paper

    Michael Rotman, Amit Dekel, Ran Ilan Ber, Lior Wolf, and Yaron Oz.

13. Non-equispaced Fourier neural solvers for PDEs. arXiv, 2022. paper

    Haitao Lin, Lirong Wu, Yongjie Xu, Yufei Huang, Siyuan Li, Guojiang Zhao, Stan Z, and Li Cari.

14. Incremental spectral learning Fourier neural operator. arXiv, 2022. paper

    Jiawei Zhao, Robert Joseph George, Yifei Zhang, Zongyi Li, and Anima Anandkumar.

15. Fourier continuation for exact derivative computation in physics-informed neural operators. arXiv, 2022. paper

    Haydn Maust, Zongyi Li, Yixuan Wang, Daniel Leibovici, Oscar Bruno, Thomas Hou, and Anima Anandkumar.

16. Non-equispaced Fourier neural solvers for PDEs. arXiv, 2023. paper

    Haitao Lin, Lirong Wu, Yongjie Xu, Yufei Huang, Siyuan Li, Guojiang Zhao, Stan Z, and Li Cari.

17. Learning-based solutions to nonlinear hyperbolic PDEs: Empirical insights on generalization errors. arXiv, 2023. paper

    Bilal Thonnam Thodi, Sai Venkata Ramana Ambadipudi, and Saif Eddin Jabari.

18. Domain agnostic Fourier neural operators. arXiv, 2023. paper

    Ning Liu, Siavash Jafarzadeh, and Yue Yu.

19. Spherical Fourier neural operators: Learning stable dynamics on the sphere. ICML, 2023. paper

    Boris Bonev, Thorsten Kurth, Christian Hundt, Jaideep Pathak, Maximilian Baust, Karthik Kashinath, and Anima Anandkumar.

20. Group equivariant Fourier neural operators for partial differential equations. ICML, 2023. paper

    Jacob Helwig, Xuan Zhang, Cong Fu, Jerry Kurtin, Stephan Wojtowytsch, and Shuiwang Ji.

21. Speeding up Fourier neural operators via mixed precision. arXiv, 2023. paper

    Colin White, Renbo Tu, Jean Kossaifi, Gennady Pekhimenko, Kamyar Azizzadenesheli, and Anima Anandkumar.

22. Geometry-informed neural operator for large-scale 3D PDEs. arXiv, 2023. paper

    Zongyi Li, Nikola Borislavov Kovachki, Chris Choy, Boyi Li, Jean Kossaifi, Shourya Prakash Otta, Mohammad Amin Nabian, Maximilian Stadler, Christian Hundt, Kamyar Azizzadenesheli, and Anima Anandkumar.

23. Deep equilibrium based neural operators for steady-state PDEs. NIPS, 2023. paper

    Tanya Marwah, Ashwini Pokle, J Zico Kolter, Zachary Chase Lipton, Jianfeng Lu, and Andrej Risteski.

24. A born fourier neural operator for solving Poisson’s equation with limited data and arbitrary domain deformation. TAP, 2023. paper

    Zheng Zong, Yusong Wang, Siyuan He, and Zhun Wei.

25. Approximating numerical flux by Fourier neural operators for the hyperbolic conservation laws. arXiv, 2024. paper

    Taeyoung Kim and Myungjoo Sang.

Graph Network

1. Message passing neural PDE solvers. ICLR, 2022. paper

   Johannes Brandstetter, Daniel E. Worrall, and Max Welling.

2. Predicting physics in mesh-reduced space with temporal attention. ICLR, 2022. paper

   Xu Han, Han Gao, Tobias Pfaff, Jianxun Wang, and Liping Liu.

3. Learning mesh-based simulation with graph networks. ICLR, 2021. paper

   Tobias Pfaff, Meire Fortunato, Alvaro Sanchez-Gonzalez, and Peter Battaglia.

4. Learning large-scale subsurface simulations with a hybrid graph network simulator. KDD, 2022. paper

   Tailin Wu, Qinchen Wang, Yinan Zhang, Rex Ying, Kaidi Cao, Rok Sosič, Ridwan Jalali, Hassan Hamam, Marko Maucec, and Jure Leskovec.

5. Unravelling the performance of physics-informed graph neural networks for dynamical systems. NIPS, 2022. paper

   Abishek Thangamuthu, Gunjan Kumar, Suresh Bishnoi, Ravinder Bhattoo, N M Anoop Krishnan, and Sayan Ranu.

6. Learning the solution operator of boundary value problems using graph neural networks. ICML, 2022. paper

   Winfried Lötzsch, Simon Ohler, and Johannes S. Otterbach.

7. Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problems. Computer Methods in Applied Mechanics and Engineering, 2022. paper

   Han Gao, Matthew J.Zahr, and Jianxun Wang.

8. Modular flows: Differential molecular generation. NIPS, 2022. paper

   Yogesh Verma, Samuel Kaski, Markus Heinonen, and Vikas Garg.

9. Learning to solve PDE-constrained inverse problems with graph networks. ICML, 2022. paper

   Zhao Qingqing, David B. Lindell, and Gordon Wetzstein.

10. Physics-embedded neural networks: E(n)-equivariant graph neural PDE solvers. NIPS, 2022. paper

    Masanobu Horie and Naoto Mitsume.

11. PDE-GCN: Novel architectures for graph neural networks motivated by partial differential equations. ICLR, 2021. paper

    Moshe Eliasof, Eldad Haber, and Eran Treister.

12. Physics-aware difference graph networks for sparsely-observed dynamics. ICLR, 2020. paper

    Sungyong Seo, Chuizheng Meng, and Yan Liu.

13. Combining differentiable PDE solvers and graph neural networks for fluid flow prediction. ICML, 2022. paper

    Filipe de Avila Belbute-Peres, Thomas D. Economon, and J. Zico Kolter.

14. Learning continuous-time PDEs from sparse data with graph neural networks. ICLR, 2021. paper

    Valerii Iakovlev, Markus Heinonen, and Harri Lähdesmäki.

15. Learning to simulate complex physics with graph networks. ICML, 2020. paper

    Alvaro Sanchez-Gonzalez, Jonathan Godwin, Tobias Pfaff, Rex Ying, Jure Leskovec, and Peter W. Battaglia.

16. Multi-scale physical representations for approximating PDE solutions with graph neural operators. ICLR, 2022. paper

    Léon Migus, Yuan Yin, Jocelyn Ahmed Mazari, and Patrick Gallinari.

17. DS-GPS: A deep statistical graph Poisson solver (for faster CFD simulations). NIPS, 2022. paper

    Matthieu Nastorg, Marc Schoenauer, Guillaume Charpiat, Thibault Faney, Jean-Marc Gratien, and Michele-Alessandro Bucci.

18. PF-GNN: Differentiable particle filtering based approximation of universal graph representations. ICLR, 2022. paper

    Mohammed Haroon Dupty, Yanfei Dong, and Wee Sun Lee.

19. GRAND: Graph neural diffusion. ICML, 2021. paper

    Benjamin Paul Chamberlain, James Rowbottom, Maria I. Gorinova, Stefan D Webb, Emanuele Rossi, and Michael M. Bronstein.

20. GRAND++: Graph neural diffusion with a source term. ICML, 2022. paper

    Matthew Thorpe, Tan Minh Nguyen, Hedi Xia, Thomas Strohmer, Andrea Bertozzi, Stanley Osher, and Bao Wang.

21. Neural networks trained to solve differential equations learn general representations. NIPS, 2018. paper

    Martin Magill, Faisal Qureshi, and Hendrick de Haan.

22. Graph element networks: Adaptive, structured computation and memory. ICML, 2019. paper

    Ferran Alet, Adarsh Keshav Jeewajee, Maria Bauza Villalonga, Alberto Rodriguez, Tomas Lozano-Perez, and Leslie Kaelbling.

23. Physics-constrained unsupervised learning of partial differential equations using meshes. arXiv, 2022. paper

    Mike Y. Michelis and Robert K. Katzschmann.

24. Neural PDE solvers for irregular domains. arXiv, 2022. paper

    Biswajit Khara, Ethan Herron, Zhanhong Jiang, Aditya Balu, Chih-Hsuan Yang, Kumar Saurabh, Anushrut Jignasu, Soumik Sarkar, Chinmay Hegde, Adarsh Krishnamurthy, and Baskar Ganapathysubramanian.

25. Bi-stride multi-scale graph neural network for mesh-based physical simulation. arXiv, 2022. paper

    Yadi Cao, Menglei Chai, Minchen Li, and Chenfanfu Jiang.

26. Learning time-dependent PDE solver using message passing graph neural networks. arXiv, 2022. paper

    Pourya Pilva and Ahmad Zareei.

27. On the robustness of graph neural diffusion to topology perturbations. arXiv, 2022. paper

    Yang Song, Qiyu Kang, Sijie Wang, Zhao Kai, and Wee Peng Tay.

28. STONet: A neural-operator-driven spatio-temporal network. arXiv, 2022. paper

    Haitao Lin, Guojiang Zhao, Lirong Wu, and Stan Z. Li.

29. PhyGNNet: Solving spatiotemporal PDEs with physics-informed graph neural network. arXiv, 2022. paper

    Longxiang Jiang, Liyuan Wang, Xinkun Chu, Yonghao Xiao, and Hao Zhang.

30. Differentiable physics-informed graph networks. arXiv, 2019. paper

    Sungyong Seo and Yan Liu.

31. MG-GNN: Multigrid graph neural networks for learning multilevel domain decomposition methods. ICML, 2023. paper

    Ali Taghibakhshi, Nicolas Nytko, Tareq Uz Zaman, Scott MacLachlan, Luke Olson, and Matthew West.

32. Multi-scale message passing neural PDE solvers. arXiv, 2023. paper

    Léonard Equer, T. Konstantin Rusch, and Siddhartha Mishra.

33. An implicit GNN solver for Poisson-like problems. arXiv, 2023. paper

    Matthieu Nastorg, Michele-Alessandro Bucci, Thibault Faney, Jean-Marc Gratien, Guillaume Charpiat, and Marc Schoenauer.

34. GNN-based physics solver for time-independent PDEs. arXiv, 2023. paper

    Rini Jasmine Gladstone, Helia Rahmani, Vishvas Suryakumar, Hadi Meidani, Marta D'Elia, and Ahmad Zareei.

35. E(3) equivariant graph neural networks for particle-based fluid mechanics. ICLR, 2023. paper

    Artur P. Toshev, Gianluca Galletti, Johannes Brandstetter, Stefan Adami, and Nikolaus A. Adams.

36. Long-short-range message-passing: A physics-informed framework to capture non-local interaction for scalable molecular dynamics simulation. arXiv, 2023. paper

    Yunyang Li, Yusong Wang, Lin Huang, Han Yang, Xinran Wei, Jia Zhang, Tong Wang, Zun Wang, Bin Shao, and Tieyan Liu.

37. A graph convolutional autoencoder approach to model order reduction for parametrized PDEs. arXiv, 2023. paper

    Federico Pichi, Beatriz Moya, and Jan S. Hesthaven.

38. GAD-NR: Graph anomaly detection via neighborhood reconstruction. arXiv, 2023. paper

    Amit Roy, Juan Shu, Jia Li, Carl Yang, Olivier Elshocht, Jeroen Smeets, and Pan Li.

39. GPINN: Physics-informed neural network with graph embedding. arXiv, 2023. paper

    Yuyang Miao and Haolin Li.

40. GNRK: Graph Neural Runge-Kutta method for solving partial differential equations. arXiv, 2023. paper

    Hoyun Choi, Sungyeop Lee, B. Kahng, and Junghyo Jo.

41. Equivariant graph neural operator for modeling 3D dynamics. arXiv, 2024. paper

    Minkai Xu, Jiaqi Han, Aaron Lou, Jean Kossaifi, Arvind Ramanathan, Kamyar Azizzadenesheli, Jure Leskovec, Stefano Ermon, and Anima Anandkumar.

Green Function

1. Learning Green's functions associated with time-dependent partial differential equations. JMLR, 2022. paper

   Nicolas Boullé, Seick Kim, Tianyi Shi, and Alex Townsend.

2. BI-GreenNet: Learning Green's functions by boundary integral network. arXiv, 2022. paper

   Guochang Lin, Fukai Chen, Pipi Hu, Xiang Chen, Junqing Chen, Jun Wang, and Zuoqiang Shi.

3. DeepGreen: Deep learning of Green’s functions for nonlinear boundary value problems. Scientific Reports, 2021. paper

   Craig R. Gin, Daniel E. Shea, Steven L. Brunton, and J. Nathan Kutz.

4. Data-driven discovery of Green’s functions with human-understandable deep learning. Scientific Reports, 2022. paper

   Nicolas Boullé, Christopher J. Earls, and Alex Townsend.

5. Data-driven discovery of Green's functions. Doctoral Dissertation, 2022. Ph.D.

   Nicolas Boullé.

6. Principled interpolation of Green's functions learned from data. arXiv, 2022. paper

   Harshwardhan Praveen, Nicolas Boulle, and Christopher Earls.

7. Deep generalized Green's functions. arXiv, 2023. paper

   Rixi Peng, Juncheng Dong, Jordan Malof, Willie J. Padilla, Vahid Tarokh.

8. Operator approximation of the wave equation based on deep learning of Green’s function. arXiv, 2023. paper

   Ziad Aldirany, R´egis Cottereau, Marc Laforest, and Serge Prudhomme.

9. Deep surrogate model for learning Green's function associated with linear reaction-diffusion operator. arXiv, 2023. paper

   Junqing Ji, Lili Ju, and Xiaoping Zhang.

Finite Element

1. Composing partial differential equations with physics-aware neural networks. ICML, 2022. paper

   Matthias Karlbauer, Timothy Praditia, Sebastian Otte, Sergey Oladyshkin, and Wolfgang Nowak.

2. Learning the dynamics of physical systems from sparse observations with finite element networks. ICLR, 2022. paper

   Marten Lienen and Stephan Günnemann.

3. A unified hard-constraint framework for solving geometrically complex PDEs. NIPS, 2022. paper

   Songming Liu, Zhongkai Hao, Chengyang Ying, Hang Su, Jun Zhu, and Ze Cheng.

4. LSH-SMILE: Locality sensitive Hashing accelerated simulation and learning. NIPS, 2021. paper

   Chonghao Sima and Yexiang Xue.

5. Amortized finite element analysis for fast PDE-constrained optimization. ICML, 2020. paper

   Tianju Xue, Alex Beatson, Sigrid Adriaenssens, and Ryan Adams.

6. Learning neural PDE solvers with convergence guarantees. ICLR, 2019. paper

   Jun-Ting Hsieh, Shengjia Zhao, Stephan Eismann, Lucia Mirabella, and Stefano Ermon.

7. Hybrid finite difference with the physics-informed neural network for solving PDE in complex geometries. arXiv, 2022. paper

   Zixue Xiang, Wei Peng, Weien Zhou, and Wen Yao.

8. Multilayer perceptron-based surrogate models for finite element analysis. arXiv, 2022. paper

   Lawson Oliveira Lima, Julien Rosenberger, Esteban Antier, and Frederic Magoules.

9. Integration of physics-informed operator learning and finite element method for parametric learning of partial differential equations. arXiv, 2024. paper

   Shahed Rezaei, Ahmad Moeineddin, Michael Kaliske, and Markus Apel.

10. Error assessment of an adaptive finite elements—neural networks method for an elliptic parametric PDE. Computer Methods in Applied Mechanics and Engineering, 2024. paper

    Alexandre Caboussat, Maude Girardin, and Marco Picasso.

Convolution

1. PDE-Net: Learning PDEs from data. ICML, 2018. paper

   Zichao Long, Yiping Lu, Xianzhong Ma, and Bin Dong.

2. PDE-Net 2.0: Learning PDEs from data with a numeric-symbolic hybrid deep network. JCP, 2019. paper

   Zichao Long, Yiping Lu, and Bin Dong.

3. Physics-informed CNNs for super-resolution of sparse observations on dynamical systems. NIPS, 2022. paper

   Daniel Kelshaw, Georgios Rigas, and Luca Magri.

4. Deep-pretrained-FWI: Combining supervised learning with physics-informed neural network. NIPS, 2022. paper

   Ana Paula O. Muller, Clecio R. Bom, Jessé C. Costa, Matheus Klatt, Elisângela L. Faria, Marcelo P. de Albuquerque, and Márcio P. de Albuquerque.

5. Learning time-dependent PDEs with a linear and nonlinear separate convolutional neural network. JCP, 2022. paper

   Jiagang Qu, Weihua Cai, and YijunZhao.

6. PhyCRNet: Physics-informed convolutional-recurrent network for solving spatiotemporal PDEs. JCP, 2022. paper

   Pu Ren, Chengping Rao, Yang Liu, Jianxun Wang, and Hao Sun.

7. Spline-PINN: Approaching PDEs without data using fast, physics-informed Hermite-Spline CNNs. AAAI, 2019. paper

   Nils Wandel, Michael Weinmann, Michael Neidlin, and Reinhard Klein.

8. Deep convolutional Ritz method: Parametric PDE surrogates without labeled data. arXiv, 2022. paper

   Jan Niklas Fuhg, Arnav Karmarkar, Teeratorn Kadeethum, Hongkyu Yoon, and Nikolaos Bouklas.

9. Deep convolutional architectures for extrapolative forecasts in time-dependent flow problems. arXiv, 2022. paper

   Pratyush Bhatt, Yash Kumar, and Azzeddine Soulaimani.

10. Phase2Vec: Dynamical systems embedding with a physics-informed convolutional network. arXiv, 2022. paper

    Matthew Ricci, Noa Moriel, Zoe Piran, and Mor Nitzan.

11. Numerical approximation based on deep convolutional neural network for high-dimensional fully nonlinear merged PDEs and 2BSDEs. arXiv, 2022. paper

    Xu Xiao and Wenlin Qiu.

12. SplineCNN: Fast geometric deep learning with continuous B-spline kernels. CVPR, 2018. paper

    Matthias Fey, Jan Eric Lenssen, Frank Weichert, and Heinrich Muller.

13. Physics-informed deep super-resolution for spatiotemporal data. arXiv, 2022. paper

    Pu Ren, Chengping Rao, Yang Liu, Zihan Ma, Qi Wang, Jianxun Wang, and Hao Sun.

14. MeshFreeFLowNet: A physics-constrained deep continuous space-time super-resolution framework. SC20, 2020. paper

    Chiyu Max Jiang, Soheil Esmaeilzadeh, Kamyar Azizzadenesheli, Karthik Kashinath, Mustafa Mustafa, Hamdi A. Tchelepi, Philip Marcus Prabhat, and Anima Anandkumar.

15. Convolutional neural operators. arXiv, 2023. paper

    Bogdan Raonić, Roberto Molinaro, Tobias Rohner, Siddhartha Mishra, and Emmanuel de Bezenac.

16. An unsupervised latent/output physics-informed convolutional-LSTM network for solving partial differential equations using peridynamic differential operator. Computer Methods in Applied Mechanics and Engineering, 2023. paper

    Arda Mavi, Ali Can Bekar, Ehsan Haghigh, and Erdogan Madenci.

17. Clifford neural layers for PDE modeling. ICLR, 2023. paper

    Johannes Brandstetter, Rianne van den Berg, Max Welling, and Jayesh K Gupta.

18. Neural partial differential equations with functional convolution. arXiv, 2023. paper

    Ziqian Wu, Xingzhe He, Yijun Li, Cheng Yang, Rui Liu, Shiying Xiong, and Bo Zhu.

19. Multilevel CNNs for parametric PDEs. arXiv, 2023. paper

    Cosmas Heiß, Ingo Gühring, and Martin Eigel.

20. Encoding physics to learn reaction–diffusion processes. NMI, 2023. paper

    Chengping Rao, Pu Ren, Qi Wang, Oral Buyukozturk, Hao Sun, and Yang Liu.

21. PhySR: Physics-informed deep super-resolution for spatiotemporal data. JCP, 2023. paper

    Pu Ren, Chengping Rao, Yang Liu, Zihan Ma, Qi Wang, Jianxun Wang, and Hao Sun.

22. Local convolution enhanced global Fourier neural operator for multiscale dynamic spaces prediction. arXiv, 2023. paper

    Xuanle Zhao, Yue Sun, Tielin Zhang, and Bo Xu.

23. A practical existence theorem for reduced order models based on convolutional autoencoders. arXiv, 2024. paper

    Nicola Rares Franco and Simone Brugiapaglia.

AutoEncoder

1. Integral autoencoder network for discretization-invariant learning. JMLR, 2022. paper

   Yong Zheng Ong, Zuowei Shen, and Haizhao Yang.

2. Variational autoencoding neural operators. arXiv, 2023. paper

   Jacob H. Seidman, Georgios Kissas, George J. Pappas, and Paris Perdikaris.

3. PI-VEGAN: Physics informed variational embedding generative adversarial networks for stochastic differential equations. arXiv, 2023. paper

   Ruisong Gao, Yufeng Wang, Min Yang, and Chuanjun Chen.

4. On the latent dimension of deep autoencoders for reduced order modeling of PDEs parametrized by random fields. arXiv, 2023. paper

   Nicola Rares Franco, Daniel Fraulin, Andrea Manzoni, and Paolo Zunino.

5. Modeling unknown stochastic dynamical system via Autoencoder. arXiv, 2023. paper

   Zhongshu Xu, Yuan Chen, Qifan Chen, and Dongbin Xiu.

Neural Operator

1. Multiwavelet-based operator learning for differential equations. NIPS, 2021. paper

   Gaurav Gupta, Xiongye Xiao, and Paul Bogdan.

2. On the representation of solutions to elliptic PDEs in Barron space. NIPS, 2021. paper

   Ziang Chen, Jianfeng Lu, and Yulong Lu.

3. Low-rank registration based manifolds for convection-dominated PDEs. AAAI, 2021. paper

   Rambod Mojgani and Maciej Balajewicz.

4. Isogeometric neural networks: A new deep learning approach for solving parameterized partial differential equations. Computer Methods in Applied Mechanics and Engineering, 2023. paper

   Joshua Gasick and Xiaoping Qian.

5. A nonlocal physics-informed deep learning framework using the peridynamic differential operator. Computer Methods in Applied Mechanics and Engineering, 2021. paper

   Ehsan Haghighat, Ali CanBekar, Erdogan Madenci, and Ruben Juanes.

6. Kernel flows: From learning kernels from data into the abyss. JCP, 2019. paper

   Houman Owhadi and Gene Ryan Yoo.

7. On neurosymbolic solutions for PDEs. arXiv, 2022. paper

   Ritam Majumdar, Vishal Jadhav, Anirudh Deodhar, Shirish Karande, and Lovekesh Vig.

8. An introduction to kernel and operator learning methods for homogenization by self-consistent clustering analysis. arXiv, 2022. paper

   Owen Huang, Sourav Saha, Jiachen Guo, and Wing Kam Liu.

9. Nonparametric learning of kernels in nonlocal operators. arXiv, 2022. paper

   Fei Lu, Qingci An, and Yue Yu.

10. Spectral neural operators. arXiv, 2022. paper

    V. Fanasko and I. Oseledets.

11. NOMAD: Nonlinear manifold decoders for operator learning. arXiv, 2022. paper

    Jacob H. Seidman, Georgios Kissas, Paris Perdikaris, and George J. Pappas.

12. U-NO: U-shaped neural operators. arXiv, 2022. paper

    Md Ashiqur Rahman, Zachary E. Ross, and Kamyar Azizzadenesheli.

13. Wavelet neural operator: A neural operator for parametric partial differential equations. arXiv, 2022. paper

    Tapas Tripura and Souvik Chakraborty.

14. Pseudo-differential integral operator for learning solution operators of partial differential equations. arXiv, 2022. paper

    Jin Young Shin, Jae Yong Lee, and Hyung Ju Hwang.

15. Nonlinear reconstruction for operator learning of PDEs with discontinuities. arXiv, 2022. paper

    Samuel Lanthaler, Roberto Molinaro, Patrik Hadorn, and Siddhartha Mishra.

16. GeONet: A neural operator for learning the Wasserstein geodesic. arXiv, 2022. paper

    Andrew Gracyk and Xiaohui Chen.

17. Render unto numerics: Orthogonal polynomial neural operator for PDEs with non-periodic boundary conditions. arXiv, 2022. paper

    Ziyuan Liu, Haifeng Wang, Kaijuna Bao, Xu Qian, Hong Zhang, and Songhe Song.

18. DOSNet as a non-black-box PDE solver: When deep learning meets operator splitting. arXiv, 2022. paper

    Yuan Lan, Zhen Li, Jie Sun, and Yang Xiang.

19. Guiding continuous operator learning through physics-based boundary constraints. arXiv, 2022. paper

    Nadim Saad, Gaurav Gupta, Shima Alizadeh, and Danielle C. Maddix.

20. BelNet: Basis enhanced learning, a mesh-free neural operator. arXiv, 2022. paper

    Zecheng Zhang, Wing Tat Leung, and Hayden Schaeffer.

21. INO: Invariant neural operators for learning complex physical systems with momentum conservation. arXiv, 2022. paper

    Ning Liu, Yue Yu, Huaiqian You, and Neeraj Tatikola.

22. BINN: A deep learning approach for computational mechanics problems based on boundary integral equations. arXiv, 2023. paper

    Jia Sun, Yinghua Liu, Yizheng Wang, Zhenhan Yao, and Xiaoping Zheng.

23. Koopman neural operator as a mesh-free solver of non-linear partial differential equations. arXiv, 2023. paper

    Wei Xiong, Xiaomeng Huang, Ziyang Zhang, Ruixuan Deng, Pei Sun, and Yang Tian.

24. Physics-informed Koopman network. arXiv, 2022. paper

    Yuying Liu, Aleksei Sholokhov, Hassan Mansour, and Saleh Nabi.

25. Deep operator learning lessens the curse of dimensionality for PDEs. arXiv, 2023. paper

    Ke Chen, Chunmei Wang, and Haizhao Yang.

26. Algorithmically designed artificial neural networks (ADANNs): Higher order deep operator learning for parametric partial differential equations. arXiv, 2023. paper

    Arnulf Jentzen, Adrian Riekert, and Philippe von Wurstemberger.

27. Entropy-dissipation informed neural network for McKean-Vlasov type. arXiv, 2023. paper

    Zebang Shen and Zhenfu Wang.

28. A neural PDE solver with temporal stencil modeling. ICML, 2023. paper

    Zhiqing Sun, Yiming Yang, and Shinjae Yoo.

29. Neural operator learning for long-time integration in dynamical systems with recurrent neural networks. arXiv, 2023. paper

    Katarzyna Michałowska, Somdatta Goswami, George Em Karniadakis, and Signe Riemer-Sørensen.

30. Coupled multiwavelet operator learning for coupled differential equations. ICLR, 2023. paper

    Xiongye Xiao, Defu Cao, Ruochen Yang, Gaurav Gupta, Chenzhong Yin, Gengshuo Liu, Radu Balan, and Paul Bogdan.

31. LNO: Laplace neural operator for solving differential equations. arXiv, 2023. paper

    Qianying Cao, Somdatta Goswami, and George Em Karniadakis.

32. Vandermonde neural operator. arXiv, 2023. paper

    Levi Lingsch, Mike Michelis, Sirani M. Perera, Robert K. Katzschmann, and Siddartha Mishra.

33. Energy-dissipative evolutionary deep operator neural networks. arXiv, 2023. paper

    Jiahao Zhang, Shiheng Zhang, Jie Shen, and Guang Lin.

34. Physics informed WNO. arXiv, 2023. paper

    Navaneeth N, Tapas Tripura, and Souvik Chakraborty.

35. Corrector operator to enhance accuracy and reliability of neural operator surrogates of nonlinear variational boundary-value problems. arXiv, 2023. paper

    Prashant K. Jha and J. Tinsley Oden.

36. HNO: Hyena neural operator for solving PDEs. arXiv, 2023. paper

    Saurabh Patil, Zijie Li, and Amir Barati Farimani.

37. Finite element operator network for solving parametric PDEs. arXiv, 2023. paper

    Jae Yong Lee, Seungchan Ko, and Youngjoon Hong.

38. PDE-Refiner: Achieving accurate long rollouts with neural PDE solvers. arXiv, 2023. paper

    Phillip Lippe, Bastiaan S. Veeling, Paris Perdikaris, Richard E. Turner, and Johannes Brandstetter.

39. Online infinite-dimensional regression: Learning linear operators. arXiv, 2023. paper

    Vinod Raman, Unique Subedi, and Ambuj Tewari.

40. Online infinite-dimensional regression: Learning linear operators. arXiv, 2023. paper

    Vinod Raman, Unique Subedi, and Ambuj Tewari.

41. Deep-OSG: Deep learning of operators in semigroup. JCP, 2023. paper

    Junfeng Chen and Kailiang Wu.

42. D2NO: Efficient handling of heterogeneous input function spaces with distributed deep neural operators. arXiv, 2023. paper

    Zecheng Zhang, Christian Moya, Lu Lu, Guang Lin, and Hayden Schaeffer.

43. Physics informed WNO. Computer Methods in Applied Mechanics and Engineering, 2023. paper

    Navaneeth N., Tapas Tripura, and Souvik Chakraborty.

44. Neural oscillators for generalizing parametric PDEs. NIPS, 2023. paper

    Taniya Kapoor, Abhishek Chandra, and Daniel M. Tartakovsky.

45. B-LSTM-MIONet: Bayesian LSTM-based neural operators for learning the response of complex dynamical systems to length-variant multiple input functions. arXiv, 2023. paper

    Zhihao Kong, Amirhossein Mollaali, Christian Moya, Na Lu, and Guang Lin.

46. GIT-Net: Generalized integral transform for operator learning. arXiv, 2023. paper

    Chao Wang and Alexandre Hoang Thiery.

47. Neural spectral methods: Self-supervised learning in the spectral domain. arXiv, 2023. paper

    Yiheng Du, Nithin Chalapathi, and Aditi S. Krishnapriyan.

48. A nonlinear-manifold reduced-order model and operator learning for partial differential equations with sharp solution gradients. Computer Methods in Applied Mechanics and Engineering, 2023. paper

    Peiyi Chen, Tianchen Hu, and Johann Guilleminot.

49. Operator-learning-inspired modeling of neural ordinary differential equations. arXiv, 2023. paper

    Woojin Cho, Seunghyeon Cho, Hyundong Jin, Jinsung Jeon, Kookjin Lee, Sanghyun Hong, Dongeun Lee, Jonghyun Choi, and Noseong Park.

50. Operator learning for hyperbolic partial differential equations. arXiv, 2023. paper

    Christopher Wang and Alex Townsend.

51. BENO: Boundary-embedded neural operators for elliptic PDEs. ICLR, 2024. paper

    Haixin Wang, Jiaxin Li, Anubhav Dwivedi, Kentaro Hara, and Tailin Wu.

52. Koopman operator learning using invertible neural networks. JCP, 2024. paper

    Yuhuang Meng, Jianguo Huang, and Yue Qiu.

Machine Learning

1. Machine learning–accelerated computational fluid dynamics. PNAS, 2021. paper

   Dmitrii Kochkov, Jamie A. Smith, Ayya Alieva, and Stephan Hoyer.

2. Learning data-driven discretization for partial differential equations. PNAS, 2019. paper

   Yohai Bar-Sinai, Stephan Hoyer, Jason Hickey, and Michael P. Brenner.

3. Solving high-dimensional partial differential equations using deep learning. PNAS, 2018. paper

   Jiequn Han, Arnulf Jentzen, and Weinan E.

4. Enhancing computational fluid dynamics with machine learning. NCS, 2022. paper

   Ricardo Vinuesa and Steven L. Brunton.

5. Solving high-dimensional parabolic PDEs using the tensor train format. ICML, 2021. paper

   Lorenz Richter, Leon Sallandt, and Nikolas Nüsken.

6. A hybrid reduced basis and machine-learning algorithm for building surrogate models: A first application to electromagnetism. NIPS, 2022. paper

   Alejandro Ribés and Ruben Persicot.

7. Numerically solving parametric families of high-dimensional Kolmogorov partial differential equations via deep learning. NIPS, 2020. paper

   Julius Berner, Markus Dablander, and Philipp Grohs.

8. Nonlocal kernel network (NKN): A stable and resolution-independent deep neural network. JCP, 2022. paper

   Huaiqian You, Yue Yu, Marta D'Elia, Tian Gao, and Stewart Silling.

9. On computing the hyperparameter of extreme learning machines: Algorithm and application to computational PDEs, and comparison with classical and high-order finite elements. JCP, 2022. paper

   Suchuan Dong and Jielin Yang.

10. Int-Deep: A deep learning initialized iterative method for nonlinear problems. JCP, 2022. paper

    Jianguo Huang, Haoqin Wang, and Haizhao Yang.

11. DeLISA: Deep learning based iteration scheme approximation for solving PDEs. JCP, 2022. paper

    Ying Li, Zuojia Zhou, and Shihui Ying.

12. Physics-informed machine learning for reduced-order modeling of nonlinear problems. JCP, 2021. paper

    Wenqian Chen, Qian Wang, Jan S.Hesthaven, and Chuhua Zhang.

13. SelectNet: Self-paced learning for high-dimensional partial differential equations. JCP, 2021. paper

    Yiqi Gu, Haizhao Yang, and Chao Zhou.

14. Deep least-squares methods: An unsupervised learning-based numerical method for solving elliptic PDEs. JCP, 2020. paper

    Zhiqiang Cai, Jingshuang Chen, Min Liu, and Xinyu Liu.

15. Local extreme learning machines and domain decomposition for solving linear and nonlinear partial differential equations. Computer Methods in Applied Mechanics and Engineering, 2021. paper

    Suchuan Dong and Zongwei Li.

16. Iterative surrogate model optimization (ISMO): An active learning algorithm for PDE constrained optimization with deep neural networks. Computer Methods in Applied Mechanics and Engineering, 2021. paper

    Kjetil O.Lye, Siddhartha Mishra, Deep Ray, and Praveen Chandrashekar.

17. Probabilistic learning on manifolds constrained by nonlinear partial differential equations for small datasets. Computer Methods in Applied Mechanics and Engineering, 2021. paper

    C. Soize and R. Ghanem.

18. An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications. Computer Methods in Applied Mechanics and Engineering, 2020. paper

    E. Samaniego, C. Anitescu, S.Goswami, V.M.Nguyen-Thanh, H.Guo, K.Hamdia, X.Zhuang, and T.Rabczuk.

19. Reduced order modeling for parameterized time-dependent PDEs using spatially and memory aware deep learning. JCP, 2021. paper

    Nikolaj T.Mücke, Sander M.Bohté, and Cornelis W.Oosterlee.

20. GCN-FFNN: A two-stream deep model for learning solution to partial differential equations. Neurocomputing, 2022. paper

    Onur Bilgin, Thomas Vergutz, Siamak Mehrkanoon.

21. Deep-HyROMnet: A deep learning-based operator approximation for hyper-reduction of nonlinear parametrized PDEs. Journal of Scientific Computing, 2021. paper

    Ludovica Cicci, Stefania Fresca, and Andrea Manzoni.

22. HybridNet: Integrating model-based and data-driven learning to predict evolution of dynamical systems. CoRL, 2018. paper

    Yun Long, Xueyuan She, and Saibal Mukhopadhyay.

23. A model-constrained tangent slope learning approach for dynamical systems. arXiv, 2022. paper

    Hai V. Nguyen and Tan Bui-Thanh.

24. Solving PDEs on unknown manifolds with machine learning. arXiv, 2021. paper

    Senwei Liang, Shixiao W. Jiang, John Harlim, and Haizhao Yang.

25. Model reduction and neural networks for parametric PDEs. SIAM Journal of Computational Mathematics, 2021. paper

    Kaushik Bhattacharya, Bamdad Hosseini, Nikola B. Kovachki, and Andrew M. Stuart.

26. MOD-Net: A machine learning approach via model-operator-data network for solving PDEs. Communications in Computational Physics, 2022. paper

    Shamsulhaq Basir and Inanc Senocak.

27. A deep double Ritz method (D^2RM) for solving partial differential equations using neural networks. Computer Methods in Applied Mechanics and Engineering, 2023. paper

    Carlos Uriarte, David Pardo, Ignacio Muga, and Judit Muñoz-Matute.

28. Data- and theory-guided learning of partial differential equations using simultaneous basis function approximation and parameter estimation (SNAPE). MSSP, 2023. paper

    Sutanu Bhowmick, Satish Nagarajaiah, and Anastasios Kyrillidis.

29. Invariant preservation in machine learned PDE solvers via error correction. arXiv, 2023. paper

    Nick McGreivy and Ammar Hakim.

30. Operator learning with PCA-Net: upper and lower complexity bounds. arXiv, 2023. paper

    Samuel Lanthaler.

31. Multiscale attention via wavelet neural operators for vision Transformers. arXiv, 2023. paper

    Anahita Nekoozadeh, Mohammad Reza Ahmadzadeh, Zahra Mardani, and Morteza Mardani.

32. Pseudo-Hamiltonian neural networks for learning partial differential equations. arXiv, 2023. paper

    Sølve Eidnes and Kjetil Olsen Lye.

33. SHoP: A deep learning framework for solving high-order partial differential equations. arXiv, 2023. paper

    Tingxiong Xiao, Runzhao Yang, Yuxiao Cheng, Jinli Suo, and Qionghai Dai.

34. A Neural RDE-based model for solving path-dependent PDEs. arXiv, 2023. paper

    Bowen Fang, Hao Ni, and Yue Wu.

35. Global universal approximation of functional input maps on weighted spaces. arXiv, 2023. paper

    Christa Cuchiero, Philipp Schmocker, and Josef Teichmann.

36. Accelerating explicit time-stepping with spatially variable time steps through machine learning. Journal of Scientific Computing, 2023. paper

    Kiera van der Sande, Natasha Flyer, and Bengt Fornberg.

37. Learning elliptic partial differential equations with randomized linear algebra. Foundations of Computational Mathematics, 2023. paper

    Nicolas Boullé and Alex Townsend .

38. A deep double Ritz method (D^2RM) for solving partial differential equations using neural networks. Computer Methods in Applied Mechanics and Engineering, 2023. paper

    Carlos Uriarte, David Pardo, Ignacio Mug, and Judit Muñoz-Matute.

39. A deep learning framework for solving hyperbolic partial differential equations: Part I. arXiv, 2023. paper

    Rajat Arora.

40. Fully probabilistic deep models for forward and inverse problems in parametric PDEs. JCP, 2023. paper

    Arnaud Vadeboncoeur, Ömer Deniz Akyildiz, Ieva Kazlauskaite, Mark Girolami, and Fehmi Cirak.

41. An axiomatized PDE model of deep neural networks. arXiv, 2023. paper

    Tangjun Wang, Wenqi Tao, Chenglong Bao, and Zuoqiang Shi.

42. Element learning: A systematic approach of accelerating finite element-type methods via machine learning, with applications to radiative transfer. arXiv, 2023. paper

    Shukai Du and Samuel N. Stechmann.

43. Deep multi-step mixed algorithm for high dimensional non-linear PDEs and associated BSDEs. arXiv, 2023. paper

    Daniel Bussell and Camilo Andrés García-Trillos.

44. Artificial to spiking neural networks conversion for scientific machine learning. arXiv, 2023. paper

    Qian Zhang, Chenxi Wu, Adar Kahana, Youngeun Kim, Yuhang Li, George Em Karniadakis, and Priyadarshini Panda.

45. Data generation-based operator learning for solving partial differential equations on unbounded domains. arXiv, 2023. paper

    Jihong Wang, Xin Wang, Jing Li, and Bin Liu.

46. Slow invariant manifolds of singularly perturbed systems via physics-informed machine learning. arXiv, 2023. paper

    Dimitrios G. Patsatzis, Gianluca Fabiani, Lucia Russo, and Constantinos Siettos.

47. Spectral operator learning for parametric PDEs without data reliance. arXiv, 2023. paper

    Junho Choi, Taehyun Yun, Namjung Kim, and Youngjoon Hong.

48. The deep minimizing movement scheme. JCP, 2023. paper

    Min Sue Park, Cheolhyeong Kim, Hwijae Son, and Hyung Ju Hwang .

49. Learning nonlinear integral operators via recurrent neural networks and its application in solving integro-differential equations. arXiv, 2023. paper

    Hardeep Bassi, Yuanran Zhu, Senwei Liang, Jia Yin, Cian C. Reeves, Vojtech Vlcek, and Chao Yang.

50. Adaptive deep neural networks for solving corner singular problems. Engineering Analysis with Boundary Elements, 2023. paper

    Shaojie Zeng, Yijie Liang, and Qinghui Zhang.

51. FastSVD-ML–ROM: A reduced-order modeling framework based on machine learning for real-time applications. Computer Methods in Applied Mechanics and Engineering, 2023. paper

    G.I. Drakoulas, T.V. Gortsas, G.C. Bourantas, V.N. Burganos, and D. Polyzos.

52. Nonintrusive reduced-order models for parametric partial differential equations via data-driven operator inference. SIAM Journal on Scientific Computing, 2023. paper

    Shane A. McQuarrie, Parisa Khodabakhshi, and Karen E. Willcox.

53. Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditions. Partial Differential Equations and Applications, 2023. paper

    Victor Boussange, Sebastian Becker, Arnulf Jentzen, Benno Kuckuck, and Loïc Pellissier.

54. Neural networks with kernel-weighted corrective residuals for solving partial differential equations. arXiv, 2024. paper

    Carlos Mora, Amin Yousefpour, Shirin Hosseinmardi, and Ramin Bostanabad.

55. Spectral integrated neural networks (SINNs) for solving forward and inverse dynamic problems. arXiv, 2024. paper

    Lin Qiu, Fajie Wang, Wenzhen Qu, Yan Gu, and Qinghua Qin.

56. Deep FBSDE neural networks for solving incompressible Navier-Stokes equation and Cahn-Hilliard equation. arXiv, 2024. paper

    Yangtao Deng and Qiaolin He.

Identification

1. Data-driven discovery of partial differential equations. SA, 2017. paper

   Wei Cai and Zhiqin John Xu.

2. Robust learning from noisy, incomplete, high-dimensional experimental data via physically constrained symbolic regression. NC, 2021. paper

   Patrick A. K. Reinbold, Logan M. Kageorge, Michael F. Schatz, and Roman O. Grigoriev.

3. Discovering nonlinear PDEs from scarce data with physics-encoded learning. ICLR, 2022. paper

   Chengping Rao, Pu Ren, Yang Liu, and Hao Sun.

4. Differential spectral normalization (DSN) for PDE discovery. AAAI, 2021. paper

   Chi Chiu So, Tsz On Li, Chufang Wu, and Siu Pang Yung.

5. Learning differential operators for interpretable time series modeling. KDD, 2022. paper

   Yingtao Luo, Chang Xu, Yang Liu, Weiqing Liu, Shun Zheng, and Jiang Bian.

6. Learning emergent partial differential equations in a learned emergent space. NC, 2022. paper

   Felix P. Kemeth, Tom Bertalan, Thomas Thiem, Felix Dietrich, Sung Joon Moon, Carlo R. Laing, and Ioannis G. Kevrekidis.

7. Physics-informed learning of governing equations from scarce data. NC, 2021. paper

   Zhao Chen, Yang Liu, and Hao Sun.

8. Machine learning hidden symmetries. PRL, 2022. paper

   Ziming Liu and Max Tegmark.

9. Efficient learning of sparse and decomposable PDEs using random projection. UAI, 2022. paper

   Md Nasim, Xinghang Zhang, Anter El-Azab, and Yexiang Xue.

10. Data-driven discovery of partial differential equation models with latent variables. Physical Review E, 2019. paper

    Patrick A. K. Reinbold and Roman O. Grigoriev.

11. Physics constrained learning for data-driven inverse modeling from sparse observations. Physical Review E, 2019. paper

    Kailai Xua and Eric Darve.

12. Deep neural network modeling of unknown partial differential equations in nodal space. JCP, 2022. paper

    Zhen Chen, and Victor Churchill, Kailiang Wu, and Dongbin Xiu.

13. DeepMOD: Deep learning for model discovery in noisy data. JCP, 2021. paper

    Gert-Jan Both, Subham Choudhury, Pierre Sens, and Remy Kusters.

14. Theory-guided hard constraint projection (HCP): A knowledge-based data-driven scientific machine learning method. JCP, 2021. paper

    Yuntian Chen, Dou Huang, Dongxiao Zhang, Junsheng Zeng, Nanzhe Wang, Haoran Zhang, and Jinyue Yan.

15. Deep-learning based discovery of partial differential equations in integral form from sparse and noisy data. JCP, 2021. paper

    Hao Xua, Dongxiao Zhang, and Nanzhe Wang.

16. Peridynamics enabled learning partial differential equations. JCP, 2021. paper

    Ali C.Bekar and Erdogan Madenci.

17. Data-driven deep learning of partial differential equations in modal space. JCP, 2020. paper

    Kailiang Wu and Dongbin Xiu.

18. DLGA-PDE: Discovery of PDEs with incomplete candidate library via combination of deep learning and genetic algorithm. JCP, 2020. paper

    Hao Xu, Haibin Chang, and Dongxiao Zhang.

19. Learning nonlocal constitutive models with neural networks. Computer Methods in Applied Mechanics and Engineering, 2021. paper

    Xuhui Zhou, Jiequn Han, and Heng Xiao.

20. Data-driven identification of parametric partial differential equations. SIAM Journal on Applied Dynamical Systems, 2019. paper

    Samuel Rudy, Alessandro Alla, Steven L. Brunton, and J. Nathan Kutz.

21. PDE-READ: Human-readable partial differential equation discovery using deep learning. Neural Networks, 2022. paper

    Robert Stephany and Christopher Earls.

22. Discover: Deep identification of symbolic open-form PDEs via enhanced reinforcement-learning. arXiv, 2022. paper

    Mengge Du, Yuntian Chen, and Dongxiao Zhang.

23. ModLaNets: Learning generalisable dynamics via modularity and physical inductive bias. ICML, 2022. paper

    Yupu Lu, Shijie Lin, Guanqi Chen, and Jia Pan.

24. Robust discovery of partial differential equations in complex situations. Physical Review Research, 2021. paper

    Hao Xu and Dongxiao Zhang.

25. Modeling the dynamics of PDE systems with physics-constrained deep auto-regressive networks. JCP, 2020. paper

    Nicholas Geneva and Nicholas Zabaras.

26. Deep learning and inverse discovery of polymer self-consistent field theory inspired by physics-informed neural networks. Physical Review E, 2022. paper

    Danny Lin and Hsiu-Yu Yu.

27. Data-driven solutions and parameter discovery of the Sasa–Satsuma equation via the physics-informed neural networks method. Physica D, 2022. paper

    Haotian Luo, Lei Wang, Yabin Zhang, Gui Lu, Jingjing Su, and Yinchuan Zhao.

28. Reliable extrapolation of deep neural operators informed by physics or sparse observations. Computer Methods in Applied Mechanics and Engineering, 2023. paper

    Min Zhu, Handi Zhang, Anran Jiao, George Em Karniadakis, and Lu Lu.

29. Neural operator with regularity structure for modeling dynamics driven by SPDEs. arXiv, 2022. paper

    Peiyan Hu, Qi Meng, Bingguang Chen, Shiqi Gong, Yue Wang, Wei Chen, Rongchan Zhu, Zhiming Ma, and Tieyan Liu.

30. A PINN approach to symbolic differential operator discovery with sparse data. arXiv, 2022. paper

    Lena Podina, Brydon Eastman, and Mohammad Kohandel.

31. WeakIdent: Weak formulation for identifying differential equations using narrow-fit and trimming. arXiv, 2022. paper

    Mengyi Tang, Wenjing Liao, Rachel Kuske, and Sung Ha Kang.

32. Predicting parametric spatiotemporal dynamics by multi-resolution PDE structure-preserved deep learning. arXiv, 2022. paper

    Xinyang Liu, Hao Sun, and Jianxun Wang.

33. Data-driven modeling of Landau damping by physics-informed neural networks. arXiv, 2022. paper

    Yilan Qin, Jiayu Ma, Mingle Jiang, Chuanfei Dong, Haiyang Fu, Liang Wang, Wenjie Cheng, and Yaqiu Jin.

34. Discovering governing equations in discrete systems using PINNs. arXiv, 2022. paper

    Sheikh Saqlain, Wei Zhu, Efstathios G. Charalampidis, and Panayotis G. Kevrekidis.

35. Physics-guided generative adversarial network to learn physical models. arXiv, 2023. paper

    Kazuo Yonekura.

36. PDE-Learn: Using deep learning to discover partial differential equations from noisy, limited data. arXiv, 2022. paper

    Robert Stephany and Christopher Earls.

37. PiSL: Physics-informed Spline Learning for data-driven identification of nonlinear dynamical systems. MSSP, 2023. paper

    Fangzheng Sun, Yang Liu, Qi Wang, Hao Sun.

38. Learning physical models that can respect conservation laws. arXiv, 2023. paper

    Derek Hansen, Danielle C. Maddix, Shima Alizadeh, Gaurav Gupta, and Michael W. Mahoney.

39. Data-driven discovery of intrinsic dynamics. NMI, 2022. paper

    Daniel Floryan and Michael D. Graham.

40. Symbolic regression for PDEs using pruned differentiable programs. ICLR, 2023. paper

    Ritam Majumdar, Vishal Jadhav, Anirudh Deodhar, Shirish Karande, Lovekesh Vig, and Venkataramana Runkana.

41. Symbolic physics learner: Discovering governing equations via Monte Carlo tree search. ICLR, 2023. paper

    Fangzheng Sun, Yang Liu, Jianxun Wang, Hao Sun.

42. Learning neural constitutive laws from motion observations for generalizable PDE dynamics. ICML, 2023. paper

    Pingchuan Ma, Peter Yichen Chen, Bolei Deng, Joshua B. Tenenbaum, Tao Du, Chuang Gan, and Wojciech Matusik.

43. Learning partial differential equations in reproducing kernel Hilbert spaces. JMLR, 2023. paper

    George Stepaniants.

44. Finite expression methods for discovering physical laws from data. arXiv, 2023. paper

    Zhongyi Jiang, Chunmei Wang, and Haizhao Yang.

45. Physics-informed representation learning for emergent organization in complex dynamical systems. arXiv, 2023. paper

    Adam Rupe, Karthik Kashinath, Nalini Kumar, and James P. Crutchfield.

46. TaylorPDENet: Learning PDEs from non-grid data. arXiv, 2023. paper

    Paul Heinisch, Andrzej Dulny, Anna Krause, and Andreas Hotho.

47. Learning of discrete models of variational PDEs from data. arXiv, 2023. paper

    Christian Offen and Sina Ober-Blöbaum.

48. Towards true discovery of the differential equations. arXiv, 2023. paper

    Alexander Hvatov and Roman Titov.

49. Adaptive uncertainty-guided model selection for data-driven PDE discovery. arXiv, 2023. paper

    Pongpisit Thanasutives, Takashi Morita, Masayuki Numao, and Ken-ichi Fukui.

50. Weak-PDE-LEARN: A weak form based approach to discovering pdes from noisy, limited data. arXiv, 2023. paper

    Robert Stephany and Christopher Earls.

51. Physics-constrained robust learning of open-form PDEs from limited and noisy data. arXiv, 2023. paper

    Mengge Du, Longfeng Nie, Siyu Lou, Yuntian Chenc, and Dongxiao Zhang.

52. Equation discovery with bayesian spike-and-slab priors and efficient kernels. arXiv, 2023. paper

    Da Long, Wei W. Xing, Aditi S. Krishnapriyan, Robert M. Kirby, Shandian Zhe, and Michael W. Mahoney.

53. A Bayesian framework for discovering interpretable Lagrangian of dynamical systems from data. arXiv, 2023. paper

    Tapas Tripura and Souvik Chakraborty.

54. HyperSINDy: Deep generative modeling of nonlinear stochastic governing equations. arXiv, 2023. paper

    Mozes Jacobs, Bingni W. Brunton, Steven L. Brunton, J. Nathan Kutz, and Ryan V. Raut.

55. Sparse Bayesian machine learning for the interpretable identification of nonlinear structural dynamics: Towards the experimental data-driven discovery of a quasi zero stiffness device. MSSP, 2023. paper

    Tanmoy Chatterjee, Alexander D. Shaw, Michael I. Friswell, and Hamed Haddad Khodaparast.

56. SD-PINN: Deep learning based spatially dependent PDEs recovery. IEEE Access, 2023. paper

    Ruixian Liu and Peter Gerstoft.

57. PROSE: Predicting operators and symbolic expressions using multimodal transformers. arXiv, 2023. paper

    Yuxuan Liu, Zecheng Zhang, and Hayden Schaeffer.

58. AI-Aristotle: A physics-informed framework for systems biology gray-box identification. arXiv, 2023. paper

    Nazanin Ahmadi Daryakenari, Mario De Florio, Khemraj Shukla, and George Em Karniadakis.

59. Learning-informed parameter identification in nonlinear time-dependent PDEs. Applied Mathematics & Optimization, 2023. paper

    Christian Aarset, Martin Holler, and Tram Thi Ngoc Nguyen.

60. Data-driven discovery of PDEs via the adjoint method. arXiv, 2024. paper

    Mohsen Sadr, Tony Tohme, and Kamal Youcef-Toumi.

Inverse Design

1. DPM: Physics-informed Karhunen-Loéve and neural network approximations for solving inverse differential equation problems. AAAI, 2021. paper

   Jungeun Kim, Kookjin Lee, Dongeun Lee, Sheo Yon Jin, and Noseong Park.

2. Neural inverse operators for Solving PDE inverse problems. ICML, 2023. paper

   Roberto Molinaro, Yunan Yang, Björn Engquist, and Siddhartha Mishra.

3. Physics-informed Karhunen-Loéve and neural network approximations for solving inverse differential equation problems. JCP, 2022. paper

   Jing Li and Alexandre M.Tartakovsky.

4. Solving inverse problems in stochastic models using deep neural networks and adversarial training. Computer Methods in Applied Mechanics and Engineering, 2021. paper

   Kailai Xu and Eric Darve.

5. Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing, 2021. paper

   Lu Lu, Raphaël Pestourie, Wenjie Yao, Zhicheng Wang, Francesc Verdugo, and Steven G. Johnson.

6. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. JCP, 2019. paper

   M.Raissia, P.Perdikarisb, and George Em Karniadakis.

7. GRIDS-Net: Inverse shape design and identification of scatterers via geometric regularization and physics-embedded deep learning. arXiv, 2023. paper

   Siddharth Nair, Timothy F. Walsh, Greg Pickrell, and Fabio Semperlotti.

8. Bayesian inversion with neural operator (BINO) for modeling subdiffusion: Forward and inverse problems. arXiv, 2022. paper

   Xiongbin Yan, Zhiqin John Xu, and Zheng Ma.

9. Maximum-likelihood estimators in physics-informed neural networks for high-dimensional inverse problems. arXiv, 2023. paper

   Gabriel S. Gusmão and Andrew J. Medford.

10. Solution of physics-based inverse problems using conditional generative adversarial networks with full gradient penalty. arXiv, 2023. paper

    Deep Ray, Javier Murgoitio-Esandi, Agnimitra Dasgupta, and Assad A. Oberai.

11. Entropy structure informed learning for inverse XDE problems. arXiv, 2023. paper

    Yan Jiang, Wuyue Yang, Yi Zhu, and Liu Hong.

12. Solving multiphysics-based inverse problems with learned surrogates and constraints. arXiv, 2023. paper

    Ziyi Yin, Rafael Orozco, Mathias Louboutin, and Felix J. Herrmann.

13. Enforcing continuous symmetries in physics-informed neural network for solving forward and inverse problems of partial differential equations. JCP, 2023. paper

    Zhiyong Zhang Hui Zhang, Lisheng Zhang, and Leilei Guo.

14. Learning prediction function of prior measures for statistical inverse problems of partial differential equations. arXiv, 2023. paper

    Junxiong Jia, Deyu Meng, Zongben Xu, and Fang Yao.

15. Let data talk: Data-regularized operator learning theory for inverse problems. arXiv, 2023. paper

    Ke Chen, Chunmei Wang, and Haizhao Yang.

16. Pseudo-differential integral autoencoder network for inverse PDE operators. arXiv, 2023. paper

    Ke Chen, Jasen Lai, and Chunmei Wang.

17. Adaptive operator learning for infinite-dimensional Bayesian inverse problems. arXiv, 2023. paper

    Zhiwei Gao, Liang Yan, and Tao Zhou.

18. Variational formulations of the strong formulation -- Forward and inverse modeling using isogeometric analysis and physics-informed networks. arXiv, 2023. paper

    Kent-Andre Mardal, Jarle Sogn, and Marius Zeinhofer.

19. A method for computing inverse parametric PDE problems with random-weight neural networks. JCP, 2023. paper

    Suchuan Dong and Yiran Wang.

Neural ODE

1. Neural rough differential equations for long time series. ICML, 2021. paper

   James Morrill, Cristopher Salvi, Patrick Kidger, and James Foster.

2. LEADS: Learning dynamical systems that generalize across environments. NIPS, 2021. paper

   Yuan Yin, Ibrahim Ayed, Emmanuel de Bézenac, and Nicolas Baskiotis, and Patrick Gallinari.

3. A neural ODE interpretation of transformer layers. NIPS, 2022. paper

   Yuan Yin, Ibrahim Ayed, Emmanuel de Bézenac, Nicolas Baskiotis, and Patrick Gallinari.

4. On robustness of neural ordinary differential equations. ICLR, 2020. paper

   Hanshu Yan, Jiawei Du, Vincent Y. F. Tan, and Jiashi Feng.

5. Spectrum dependent learning curves in kernel regression and wide neural networks. ICML, 2020. paper

   Blake Bordelon, Abdulkadir Canatar, and Cengiz Pehlevan.

6. Beyond finite layer neural networks: Bridging deep architectures and numerical differential equations. ICML, 2018. paper

   Yiping Lu, Aoxiao Zhong, Quanzheng Li, and Bin Dong.

7. Neural ordinary differential equations. NIPS, 2018. paper

   Ricky T. Q. Chen, Yulia Rubanova, Jesse Bettencourt, and David K. Duvenaud.

8. Hamiltonian neural networks. ICML, 2018. paper

   Sam Greydanus, Misko Dzamba, and Jason Yosinski.

9. Hamiltonian neural networks for solving equations of motion. Physical Review E, 2022. paper

   Marios Mattheaki, David Sondak, Akshunna S. Dogra, and Pavlos Protopapas.

10. Deep learning for universal linear embeddings of nonlinear dynamics. NC, 2018. paper

    Bethany Lusch, J. Nathan Kutz, and Steven L. Brunton.

11. Stochastic physics-informed neural ordinary differential equations. arXiv, 2021. paper

    Jared O'Leary, Joel A. Paulson, and Ali Mesbah.

12. A stable and scalable method for solving initial value PDEs with neural networks. ICLR, 2023. paper

    Marc Anton Finzi, Andres Potapczynski, Matthew Choptuik, and Andrew Gordon Wilson.

13. Characteristic neural ordinary differential equation. ICLR, 2023. paper

    Xingzi Xu, Ali Hasan, Khalil Elkhalil, Jie Ding, and Vahid Tarokh.

14. Efficient certified training and robustness verification of neural ODEs. ICLR, 2023. paper

    Mustafa Zeqiri, Mark Niklas Mueller, Marc Fischer, and Martin Vechev.

15. Efficient certified training and robustness verification of neural ODEs. ICLR, 2023. paper

    Mustafa Zeqiri, Mark Niklas Mueller, Marc Fischer, and Martin Vechev.

16. Physics-informed neural ODE (PINODE): embedding physics into models using collocation points. Scientific Reports, 2023. paper

    Aleksei Sholokhov, Yuying Liu, Hassan Mansour, and Saleh Nabi.

17. Data-adaptive probabilistic likelihood approximation for ordinary differential equations. arXiv, 2023. paper

    Mohan Wu and Martin Lysy.

18. Neural signature kernels as infinite-width-depth-limits of controlled ResNets. ICML, 2023. paper

    Nicola Muca Cirone, Maud Lemercier, and Cristopher Salvi.

19. AbODE: Ab initio antibody design using conjoined ODEs. ICML, 2023. paper

    Yogesh Verma, Markus Heinonen, and Vikas Garg.

20. On the forward invariance of neural ODEs. ICML, 2023. paper

    Wei Xiao, Tsun-Hsuan Wang, Ramin Hasani, Mathias Lechner, Yutong Ban, Chuang Gan, and Daniela Rus.

21. Predicting ordinary differential equations with Transformers. ICML, 2023. paper

    Sören Becker, Michal Klein, Alexander Neitz, Giambattista Parascandolo, and Niki Kilbertus.

22. Elucidating the solution space of extended reverse-time SDE for diffusion models. arXiv, 2023. paper

    Qinpeng Cui, Xinyi Zhang, Zongqing Lu, and Qingmin Liao.

23. ODE-based recurrent model-free reinforcement learning for POMDPs. arXiv, 2023. paper

    Xuanle Zhao, Duzhen Zhang, Liyuan Han, Tielin Zhang, and Bo Xu.

24. A spectral approach for learning spatiotemporal neural differential equations. arXiv, 2023. paper

    Mingtao Xia, Xiangting Li, Qijing Shen, and Tom Chou.

25. Invariant physics-informed neural networks for ordinary differential equations. arXiv, 2023. paper

    Shivam Arora, Alex Bihlo, and Francis Valiquette.

26. Stability-informed initialization of neural ordinary differential equations. arXiv, 2023. paper

    Theodor Westny, Arman Mohammadi, Daniel Jung, and Erik Frisk.

27. U^p-Net: a generic deep learning-based time stepper for parameterized spatio-temporal dynamics. Computational Mechanics, 2023. paper

    Merten Stender, Jakob Ohlsen, Hendrik Geisler, Amin Chabchoub, Norbert Hoffmann, and Alexander Schlaefer.

28. Implicit regularization of deep residual networks towards neural ODEs. ICLR, 2024. paper

    Pierre Marion, Yuhan Wu, Michael E. Sander, Gérard Biau.

Large Model

1. Prompting in-context operator learning with sensor data, equations, and natural language. arXiv, 2023. paper

   Liu Yang, Tingwei Meng, Siting Liu, and Stanley J. Osher.

2. CrunchGPT: A chatGPT assisted framework for scientific machine learning. Journal of Machine Learning for Modeling and Computing, 2023. paper

   Varun Kumar, Leonard Gleyzer, Adar Kahana, Khemraj Shukla, and George Em Karniadakis.

Mechanism

Library

1. PDEBench: An extensive benchmark for scientific machine learning. NIPS, 2022. paper

   Makoto Takamoto, Timothy Praditia, Raphael Leiteritz, Dan MacKinlay, Francesco Alesiani, Dirk Pflüger, and Mathias Niepert.

2. DeepXDE: A deep learning library for solving differential equations. SIAM Review, 2021. paper

   Lu Lu, Xuhui Meng, Zhiping Mao, and George Em Karniadakis.

3. A research framework for writing differentiable PDE discretizations in JAX. NIPS, 2021. paper

   Antonio Stanziola, Simon R. Arridge, Ben T. Cox, and Bradley E. Treeby.

4. An extensible benchmarking graph-mesh dataset for studying steady-state incompressible Navier-Stokes equations. ICLR, 2022. paper

   Florent Bonnet, Jocelyn Ahmed Mazari, Thibaut Munzer, Pierre Yser, and Patrick Gallinari.

5. PhiFlow: A differentiable PDE solving framework for deep learning via physical simulations. NIPS, 2020. paper

   Philipp Holl, Vladlen Koltun, Kiwon Um, and Nils Thuerey.

6. NVIDIA SimNet™: An AI-accelerated multi-physics simulation framework. ICCS, 2021. paper

   Oliver Hennigh, Susheela Narasimhan, Mohammad Amin Nabian, Akshay Subramaniam, Kaustubh Tangsali, Zhiwei Fang, Max Rietmann, Wonmin Byeon, and Sanjay Choudhry.

7. PyDEns: A python framework for solving differential equations with neural networks. arXiv, 2019. paper

   Alexander Koryagin, Roman Khudorozkov, and Sergey Tsimfer.

8. KoopmanLab: A PyTorch module of Koopman neural operator family for solving partial differential equations. arXiv, 2023. paper

   Wei Xiong, Muyuan Ma, Pei Sun, and Yang Tian.

9. Physics-driven machine learning models coupling PyTorch and Firedrake. arXiv, 2023. paper

   Nacime Bouziani and David A. Ham.

10. PINNacle: A comprehensive benchmark of physics-informed neural networks for solving PDEs. arXiv, 2023. paper

    Zhongkai Hao, Jiachen Yao, Chang Su, Hang Su, Ziao Wang, Fanzhi Lu, Zeyu Xia, Yichi Zhang, Songming Liu, Lu Lu, and Jun Zhu.

11. BubbleML: A multi-physics dataset and benchmarks for machine learning. arXiv, 2023. paper

    Sheikh Md Shakeel Hassan, Arthur Feeney, Akash Dhruv, Jihoon Kim, Youngjoon Suh, Jaiyoung Ryu, Yoonjin Won, and Aparna Chandramowlishwaran.

12. LagrangeBench: A Lagrangian fluid nechanics benchmarking suite. NIPS, 2023. paper

    Artur P. Toshev, Gianluca Galletti, Fabian Fritz, Stefan Adami, and Nikolaus A. Adams.

13. Zero coordinate shift: Whetted automatic differentiation for physics-informed operator learning. arXiv, 2023. paper

    Kuangdai Leng, Mallikarjun Shankar, and Jeyan Thiyagalingam.

14. CFDBench: A comprehensive benchmark for machine learning methods in fluid dynamics. arXiv, 2023. paper

    Yining Luo, Yingfa Chen, and Zhen Zhang.

15. An operator learning framework for spatiotemporal super-resolution of scientific simulations. arXiv, 2023. paper

    Valentin Duruisseaux and Amit Chakraborty.

16. ClimSim: A large multi-scale dataset for hybrid physics-ML climate emulation. NIPS, 2023. paper

    Busecke, Nora Loose, Charles I Stern, Tom Beucler, Bryce Harrop, Benjamin R Hillman, Andrea Jenney, Savannah Ferretti, Nana Liu, Anima Anandkumar, Noah D Brenowitz, Veronika Eyring, Nicholas Geneva, Pierre Gentine, Stephan Mandt, Jaideep Pathak, Akshay Subramaniam, Carl Vondrick, Rose Yu, Laure Zanna, Tian Zheng, Ryan Abernathey, Fiaz Ahmed, David C Bader, Pierre Baldi, Elizabeth Barnes, Christopher Bretherton, Peter Caldwell, Wayne Chuang, Yilun Han, Yu Huang, Fernando Iglesias-Suarez, Sanket Jantre, Karthik Kashinath, Marat Khairoutdinov, Thorsten Kurth, Nicholas Lutsko, Po-Lun Ma, Griffin Mooers, J. David Neelin, David Randall, Sara Shamekh, Mark A Taylor, Nathan Urban, Janni Yuval, Guang Zhang, and Michael Pritchard.

17. Benchmarking the robustness of neural network-based partial differential equation solvers. ICCAD, 2023. paper

    Jiaqi Gu, Mohit Dighamber, Zhengqi Gao, and Duane S. Boning.

Analysis

1. Characterizing possible failure modes in physics-informed neural networks. NIPS, 2021. paper

   Aditi Krishnapriyan, Amir Gholami, Shandian Zhe, Robert Kirby, and Michael W. Mahoney.

2. Understanding and mitigating gradient flow pathologies in physics-informed neural networks. SIAM Journal on Scientific Computing, 2021. paper

   Sifan Wang, Yujun Teng, and Paris Perdikaris.

3. When and why PINNs fail to train: A neural tangent kernel perspective. JCP, 2022. paper

   SifanWang, XinlingYu, and Paris Perdikaris.

4. When do extended physics-informed neural networks (XPINNs) improve generalization? SIAM Journal on Scientific Computing, 2022. paper

   Zheyuan Hu, Ameya D. Jagtap, George Em Karniadakis, and Kenji Kawaguchi.

5. MARS: A method for the adaptive removal of stiffness in PDEs. JCP, 2022. paper

   Laurent Duchemin and Jen Eggers.

6. Learning similarity metrics for numerical simulations. ICML, 2020. paper

   Georg Kohl, Kiwon Um, and Nils Thuerey.

7. A curriculum-training-based strategy for distributing collocation points during physics-informed neural network training. NIPS, 2021. paper

   Marcus Münzer and Chris Bard.

8. Parametric complexity bounds for approximating PDEs with neural networks. NIPS, 2021. paper

   Tanya Marwah, Zachary Lipton, and Andrej Risteski.

9. Respecting causality is all you need for training physics-informed neural networks. arXiv, 2021. paper

   Sifan Wang, Shyam Sankaran, and Paris Perdikaris.

10. CP-PINNS: Changepoints detection in PDEs using physics informed neural networks with total-variation penalty. arXiv, 2022. paper

    Zhikang Dong and Pawel Polak.

11. Stochastic projection based approach for gradient free physics informed learning. arXiv, 2022. paper

    Navaneeth N and Souvik Chakraborty.

12. Understanding the difficulty of training physics-informed neural networks on dynamical systems. arXiv, 2022. paper

    Franz M. Rohrhofer, Stefan Posch, Clemens Gößnitzer, and Bernhard C. Geiger.

13. Mitigating learning complexity in physics and equality constrained artificial neural networks. arXiv, 2022. paper

    Victor Churchill and Dongbin Xiu.

14. Improved training of physics-informed neural networks with model ensembles. arXiv, 2022. paper

    Katsiaryna Haitsiukevich and Alexander Ilin.

15. The cost-accuracy trade-off in operator learning with neural networks. arXiv, 2022. paper

    Maarten V. de Hoop, Daniel Zhengyu Huang, Elizabeth Qian, and Andrew M. Stuart.

16. Separable PINN: Mitigating the curse of dimensionality in physics-informed neural networks. NIPS, 2022. paper

    Junwoo Cho, Seungtae Nam, Hyunmo Yang, Seok-Bae Yun, Youngjoon Hong, and Eunbyung Park.

17. A curriculum-training-based strategy for distributing collocation points during physics-informed neural network training. NIPS, 2022. paper

    Marcus Münzer and Chris Bard.

18. Robustness of physics-informed neural networks to noise in sensor data. arXiv, 2022. paper

    Jian Cheng Wong, Pao-Hsiung Chiu, Chin Chun Ooi, and My Ha Da.

19. Investigations on convergence behaviour of physics informed neural networks across spectral ranges and derivative orders. arXiv, 2023. paper

    Mayank Deshpande, Siddharth Agarwal, Vukka Snigdha, and Arya Kumar Bhattacharya.

20. Stochastic projection based approach for gradient free physics informed learning. Computer Methods in Applied Mechanics and Engineering, 2023. paper

    Navaneeth N. and Souvik Chakraborty.

21. Temporal consistency loss for physics-informed neural networks. arXiv, 2023. paper

    Sukirt Thakur, Maziar Raissi, Harsa Mitra, and Arezoo Ardekani.

22. Can physics-informed neural networks beat the finite element method? arXiv, 2023. paper

    Tamara G. Grossmann, Urszula Julia Komorowska, Jonas Latz, and Carola-Bibiane Schönlieb.

23. LSA-PINN: Linear boundary connectivity loss for solving PDEs on complex geometry. arXiv, 2023. paper

    Jian Cheng Wong, Pao-Hsiung Chiu, Chinchun Ooi, and My Ha Dao, and Yew-Soon Ong.

24. On the Hyperparameters influencing a PINN’s generalization beyond the training domain. arXiv, 2023. paper

    Andrea Bonfanti, Roberto Santana, Marco Ellero, and Babak Gholami.

25. DPM: A novel training method for physics-informed neural networks in extrapolation. AAAI, 2021. paper

    Jungeun Kim, Kookjin Lee, Dongeun Lee, Sheo Yon Jhin, and Noseong Park.

26. Guiding continuous operator learning through physics-based boundary constraints. ICLR, 2023. paper

    Nadim Saad, Gaurav Gupta, Shima Alizadeh, and Danielle C. Maddix.

27. Probing optimisation in physics-informed neural networks. ICLR, 2023. paper

    Nayara Fonseca, Veronica Guidetti, and Will Trojak.

28. Nearly optimal VC-dimension and Pseudo-dimension bounds for deep neural network derivatives. arXiv, 2023. paper

    Yahong Yang, Haizhao Yang, and Yang Xiang.

29. PDE+: Enhancing generalization via PDE with adaptive distributional diffusion. arXiv, 2023. paper

    Yige Yuan, Bingbing Xu, Bo Lin, Liang Hou, Fei Sun, Huawei Shen, and Xueqi Cheng.

30. Towards foundation models for scientific machine learning: Characterizing scaling and transfer behavior. arXiv, 2023. paper

    Shashank Subramanian, Peter Harrington, Kurt Keutzer, Wahid Bhimji, Dmitriy Morozov, Michael Mahoney, and Amir Gholami.

31. Understanding and mitigating extrapolation failures in physics-informed neural networks. arXiv, 2023. paper

    Lukas Fesser, Richard Qiu, and Luca D'Amico-Wong.

32. Physics-informed neural network based on a new adaptive gradient descent algorithm for solving partial differential equations of flow problems. Physics of Fluids, 2023. paper

    Xiaojian Li, Yuhao Liu, and Zhengxian Liu.

33. The curse of dimensionality in operator learning. arXiv, 2023. paper

    Samuel Lanthaler and Andrew M. Stuart.

34. Inverse evolution layers: Physics-informed regularizers for deep neural networks. arXiv, 2023. paper

    Chaoyu Liu, Zhonghua Qiao, Chao Li, and Carola-Bibiane Schönlieb.

35. Positional embeddings for solving PDEs with evolutional deep neural networks. arXiv, 2023. paper

    Mariella Kast and Jan S Hesthaven.

36. Learning specialized activation functions for physics-informed neural networks. arXiv, 2023. paper

    Honghui Wang, Lu Lu, Shiji Song, and Gao Huang.

37. Size lowerbounds for deep operator networks. arXiv, 2023. paper

    Anirbit Mukherjee and Amartya Roy.

38. How important are specialized transforms in neural operators? arXiv, 2023. paper

    Ritam Majumdar, Shirish Karande, and Lovekesh Vig.

39. Neural oscillators for generalization of physics-informed machine learning. arXiv, 2023. paper

    Taniya Kapoor, Abhishek Chandra, Daniel M. Tartakovsky, Hongrui Wang, Alfredo Nunez, and Rolf Dollevoet.

40. Deep neural networks with ReLU, leaky ReLU, and softplus activation provably overcome the curse of dimensionality for Kolmogorov partial differential equations with Lipschitz nonlinearities in the L^p-sense. arXiv, 2023. paper

    Julia Ackermann, Arnulf Jentzen, Thomas Kruse, Benno Kuckuck, and Joshua Lee Padgett.

41. An operator preconditioning perspective on training in physics-informed machine learning. arXiv, 2023. paper

    Tim De Ryck, Florent Bonnet, Siddhartha Mishra, and Emmanuel de Bézenac.

42. A trial solution for imposing boundary conditions of partial differential equations in physics-informed neural networks. EAAI, 2023. paper

    Seyedalborz Manavi, Ehsan Fattahi, and Thomas Becker.

43. Multiple physics pretraining for physical surrogate models. arXiv, 2023. paper

    Michael McCabe, Bruno Régaldo-Saint Blancard, Liam Holden Parker, Ruben Ohana, Miles Cranmer, Alberto Bietti, Michael Eickenberg, Siavash Golkar, Geraud Krawezik, Francois Lanusse, Mariel Pettee, Tiberiu Tesileanu, Kyunghyun Cho, and Shirley Ho.

44. Lie point symmetry and physics informed networks. NIPS, 2023. paper

    Tara Akhound-Sadegh, Laurence Perreault-Levasseur, Johannes Brandstetter, Max Welling, and Siamak Ravanbakhsh.

45. Stacked networks improve physics-informed training: Applications to neural networks and deep operator networks. arXiv, 2023. paper

    Amanda A Howard, Sarah H Murphy, Shady E Ahmed, and Panos Stinis.

46. Is the neural tangent kernel of PINNs deep learning general partial differential equations always convergent? Physica D: Nonlinear Phenomena, 2023. paper

    Zijian Zhou and Zhenya Yan.

47. Can physics informed neural operators self improve? NIPS, 2023. paper

    Ritam Majumdar, Ritam_Majumdar, Amey Varhade, Shirish Karande, and Lovekesh Vig.

48. Bias-variance trade-off in physics-informed neural networks with randomized smoothing for high-dimensional PDEs. arXiv, 2023. paper

    Zheyuan Hu, Zhouhao Yang, Yezhen Wang, George Em Karniadakis, and Kenji Kawaguchi.

49. From plateaus to progress: Unveiling training dynamics of PINNs. NIPS, 2023. paper

    Rahil Pandya, Panos Parpas, and Daniel Lengyel.

50. An integrated framework for accelerating reactive flow simulation using GPU and machine learning models. arXiv, 2023. paper

    Runze Mao, Yingrui Wang, Min Zhang, Han Li, Jiayang Xu, Xinyu Dong, Yan Zhang, and Zhi X. Chen.

51. Greedy training algorithms for neural networks and applications to PDEs. JCP, 2023. paper

    Jonathan W. Siegel, Qingguo Hong, Xianlin Jin, Wenrui Hao, and Jinchao Xu.

52. How much can one learn a partial differential equation from its solution? Foundations of Computational Mathematics, 2023. paper

    Yuchen He, Hongkai Zhao, and Yimin Zhong.

53. Random feature neural networks learn black-Scholes type PDEs without curse of dimensionality, JMLR, 2023. paper

    Lukas Gonon.

54. Generalizable neural physics solvers by baldwinian evolution. arXiv, 2023. paper

    Jian Cheng Wong, Chin Chun Ooi, Abhishek Gupta, Pao-Hsiung Chiu, Joshua Shao Zheng Low, My Ha Dao, and Yew-Soon Ong.

55. Mind the gap: Constraining physics-informed neural operators to improve the quality of auto-regressive predictions. AIAA, 2024. paper

    Sean F. Current, Vilas J. Shinde, Saket Gurukar, Datta V. Gaitonde, and Srinivasan Parthasarathy.

56. Refined generalization analysis of the deep Ritz method and physics-informed neural networks. arXiv, 2024. paper

    Xianliang Xu and Zhongyi Huang.

57. PirateNets: Physics-informed deep learning with residual adaptive networks. arXiv, 2024. paper

    Sifan Wang, Bowen Li, Yuhan Chen, and Paris Perdikaris.

Domain Adaptation

1. Meta-auto-decoder for solving parametric partial differential equations. NIPS, 2022. paper

   Xiang Huang, Zhanhong Ye, Hongsheng Liu, Beiji Shi, Zidong Wang, Kang Yang, Yang Li, Bingya Weng, Min Wang, Haotian Chu, Jing Zhou, Fan Yu, Bei Hua, Lei Chen, and Bin Dong.

2. Transfer learning with physics-informed neural networks for efficient simulation of branched flows. NIPS, 2022. paper

   Raphaël Pellegrin, Blake Bullwinkel, Marios Mattheakis, and Pavlos Protopapas.

3. Identifying physical law of hamiltonian systems via meta-learning. ICLR, 2021. paper

   Seungjun Lee, Haesang Yang, and Woojae Seong.

4. One-shot learning for solution operators of partial differential equations. ICLR, 2021. paper

   Lu Lu, Haiyang He, Priya Kasimbeg, Rishikesh Ranade, and Jay Pathak.

5. A metalearning approach for physics-informed neural networks (PINNs): Application to parameterized PDEs. JCP, 2023. paper

   Michael Penwarden, Shandian Zhe, Akil Narayan, and Robert M.Kirby.

6. Meta-MgNet: Meta multigrid networks for solving parameterized partial differential equations. JCP, 2022. paper

   Yuyan Chen, Bin Dong, and Jinchao Xu.

7. Mosaic flows: A transferable deep learning framework for solving PDEs on unseen domains. Computer Methods in Applied Mechanics and Engineering, 2022. paper

   Hengjie Wang, Robert Planas, Aparna Chandramowlishwaran, and Ramin Bostanabad.

8. Meta-learning PINN loss functions. JCP, 2022. paper

   Apostolos F Psaros,Kenji Kawaguchi, and George Em Karniadakis.

9. HyperPINN: Learning parameterized differential equations with physics-informed hypernetworks. NIPS, 2021. paper

   Filipe de Avila Belbute-Peres, Yifan Chen, and Fei Sha.

10. Adversarial multi-task learning enhanced physics-informed neural networks for solving partial differential equations. IJCNN, 2022. paper

    Pongpisit Thanasutives, Masayuki Numao, and Ken-ichi Fukui.

11. Transfer physics informed neural network: A new framework for distributed physics informed neural networks via parameter sharing. Engineering with Computers, 2022. paper

    Sreehari Manikkan and Balaji Srinivasan.

12. Meta-PDE: Learning to solve PDEs quickly without a mesh. arXiv, 2022. paper

    Tian Qin, Alex Beatson, Deniz Oktay, Nick Mc Greivy, and Ryan P. Adams.

13. SVD-PINNs: Transfer learning of physics-informed neural networks via singular value decomposition. arXiv, 2022. paper

    Yihang Gao, Ka Chun Cheung, and Michael K. Ng.

14. Physics-informed neural networks (PINNs) for parameterized PDEs: A meta-learning approach. arXiv, 2021. paper

    Michael Penwarden, Shandian Zhe, Akil Narayan, and Robert M. Kirby.

15. Data-driven initialization of deep learning solvers for Hamilton-Jacobi-Bellman PDEs. arXiv, 2022. paper

    Anastasia Borovykh, Dante Kalise, Alexis Laignelet, and Panos Parpas.

16. Transfer learning based physics-informed neural networks for solving inverse problems in engineering structures under different loading scenarios. Computer Methods in Applied Mechanics and Engineering, 2023. paper

    Chen Xu, Ba Trung Cao, Yong Yuan, and Günther Meschke.

17. TransNet: Transferable neural networks for partial differential equations. arXiv, 2023. paper

    Zezhong Zhang, Feng Bao, Lili Ju, and Guannan Zhang.

18. Adaptive weighting of Bayesian physics informed neural networks for multitask and multiscale forward and inverse problems. arXiv, 2023. paper

    Sarah Perez, Suryanarayana Maddu, Ivo F. Sbalzarini, and Philippe Poncet.

19. GPT-PINN: Generative pre-trained physics-informed neural networks toward non-intrusive meta-learning of parametric PDEs. arXiv, 2023. paper

    Yanlai Chen and Shawn Koohy.

20. Gradient-enhanced physics-informed neural networks based on transfer learning for inverse problems of the variable coefficient differential equations. arXiv, 2023. paper

    Shuning Lin and Yong Chen.

21. Towards foundation models for scientific machine learning: Characterizing scaling and transfer behavior. arXiv, 2023. paper

    Shashank Subramanian, Peter Harrington, Kurt Keutzer, Wahid Bhimji, Dmitriy Morozov, Michael Mahoney, and Amir Gholami.

22. Adaptive transfer learning for PINN. JCP, 2023. paper

    Yang Liu, Wen Liu, Xunshi Yan, Shuaiqi Guo, and Chen-an Zhang.

23. Meta learning of interface conditions for multi-domain physics-informed neural networks. ICML, 2023. paper

    Shibo Li, Michael Penwarden, Yiming Xu, Conor Tillinghast, Akil Narayan, Robert Kirby, and Shandian Zhe.

24. A sequential meta-transfer (SMT) learning to combat complexities of physics-informed neural networks: Application to composites autoclave processing. arXiv, 2023. paper

    Milad Ramezankhani and Abbas S. Milani.

25. HyperLoRA for PDEs. arXiv, 2023. paper

    Ritam Majumdar, Vishal Jadhav, Anirudh Deodhar, Shirish Karande, Lovekesh Vig, and Venkataramana Runkana.

26. In-context operator learning with data prompts for differential equation problems. PNAS, 2023. paper

    Liu Yang, Siting Liu, Tingwei Meng, and Stanley J. Osher.

27. SONets: Sub-operator learning enhanced neural networks for solving parametric partial differential equations. JCP, 2023. paper

    Xiaowei Jin and Hui Li.

28. Hypernetwork-based meta-learning for low-rank physics-informed neural networks. NIPS, 2023. paper

    Woojin Cho, Kookjin Lee, Donsub Rim, and Noseong Park.

29. Meta-learning of physics-informed neural networks for efficiently solving newly given PDEs. arXiv, 2023. paper

    Tomoharu Iwata, Yusuke Tanaka, and Naonori Ueda.

30. Generalized one-shot transfer learning of linear ordinary and partial differential equations. NIPS, 2023. paper

    Tomoharu Iwata, Yusuke Tanaka, and Naonori Ueda.

31. One-shot transfer learning for nonlinear ODEs. NIPS, 2023. paper

    Wanzhou Lei, Pavlos Protopapas, and Joy Parikh.

32. PDE generalization of in-context operator networks: A study on 1D scalar nonlinear conservation laws. arXiv, 2024. paper

    Liu Yang and Stanley J. Osher.

Loss Function

1. Adaptive activation functions accelerate convergence in deep and physics-informed neural networks. JCP, 2021. paper

   Ameya D.Jagtap, KenjiKawaguchi, and George Em Karniadakis.

2. Adaptive deep neural networks methods for high-dimensional partial differential equations. JCP, 2022. paper

   Shaojie Zeng, Zong Zhang, and Qingsong Zou.

3. Self-adaptive loss balanced Physics-informed neural networks. Neurocomputing, 2022. paper

   Zixue Xiang, Wei Peng, Xu Liu, and Wen Yao.

4. Multi-objective loss balancing for physics-informed deep learning. arXiv, 2021. paper

   Rafael Bischof and Michael Kraus.

5. Self-scalable Tanh (Stan): Faster convergence and better generalization in physics-informed neural networks. arXiv, 2022. paper

   Raghav Gnanasambandam, Bo Shen, Jihoon Chung, Xubo Yue, and Zhenyu (James) Kong.

6. Is L2 physics-informed loss always suitable for training physics-informed neural network? arXiv, 2022. paper

   Chuwei Wang, Shanda Li, Di He, and Liwei Wang.

7. Loss landscape engineering via data regulation on PINNs. arXiv, 2022. paper

   Vignesh Gopakumar, Stanislas Pamela, and Debasmita Samaddar.

8. Implicit stochastic gradient descent for training physics-informed neural networks. AAAI, 2023. paper

   Ye Li, Songcan Chen, and Shengjun Huang.

9. About optimal loss function for training physics-informed neural networks under respecting causality. arXiv, 2023. paper

   Vasiliy A. Es'kin, Danil V. Davydov, Ekaterina D. Egorova, Alexey O. Malkhanov, Mikhail A. Akhukov, and Mikhail E. Smorkalov.

10. Fourier-MIONet: Fourier-enhanced multiple-input neural operators for multiphase modeling of geological carbon sequestration. arXiv, 2023. paper

    Zhongyi Jiang, Min Zhu, Dongzhuo Li, Qiuzi Li, Yanhua O. Yuan, and Lu Lu.

11. Learning from integral losses in physics informed neural networks. arXiv, 2023. paper

    Ehsan Saleh, Saba Ghaffari, Timothy Bretl, Luke Olson, and Matthew West.

12. A symmetry group based supervised learning method for solving partial differential equations. Computer Methods in Applied Mechanics and Engineering, 2023. paper

    Zhiyong Zhang, Shengjie Cai, and Hui Zhang.

Sampling

1. ADLGM: An efficient adaptive sampling deep learning Galerkin method. JCP, 2023. paper

   Andreas C.Aristotelous, Edward C.Mitchell, and Vasileios Maroulasb.

2. DMIS: Dynamic mesh-based importance sampling for training physics-informed neural networks. AAAI, 2023. paper

   Zijiang Yang, Zhongwei Qiu, and Dongmei Fu.

3. A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks. Computer Methods in Applied Mechanics and Engineering, 2023. paper

   Chenxi Wua, Min Zhua, Qinyang Tan, YadhKartha, and Lu Lu.

4. Mitigating propagation failures in PINNs using evolutionary sampling. arXiv, 2022. paper

   Arka Daw, Jie Bu, Sifan Wang, Paris Perdikaris, and Anuj Karpatne.

5. Residual-quantile adjustment for adaptive training of physics-informed neural network. arXiv, 2022. paper

   Jiayue Han, Zhiqiang Cai, Zhiyou Wu, and Xiang Zhou.

6. Adversarial sampling for solving differential equations with neural networks. NIPS, 2021. paper

   Kshitij Parwani and Pavlos Protopapas.

7. A novel adaptive causal sampling method for physics-informed neural networks. arXiv, 2022. paper

   Jia Guo, Haifeng Wang, and Chenping Hou.

8. Physics-informed neural networks with residual/gradient-based adaptive sampling methods for solving PDEs with sharp solutions. arXiv, 2023. paper

   Zhiping Mao and Xuhui Meng.

9. Active learning based sampling for high-dimensional nonlinear partial differential equations. JCP, 2023. paper

   Wenhan Gao and Chunmei Wang.

10. Adversarial adaptive sampling: Unify PINN and optimal yransport for the approximation of PDEs. ICLR, 2024. paper

    Kejun Tang, Jiayu Zhai, Xiaoliang Wan, and Chao Yang.

11. Coupling parameter and particle dynamics for adaptive sampling in neural Galerkin schemes. arXiv, 2023. paper

    Yuxiao Wen, Eric Vanden-Eijnden, and Benjamin Peherstorfer.

12. Mitigating propagation failures in physics-informed neural networks using retain-resample-release (R3) sampling. ICML, 2023. paper

    Arka Daw, Jie Bu, Sifan Wang, Paris Perdikaris, and Anuj Karpatne.

13. Good lattice training: Physics-informed neural networks accelerated by number theory. arXiv, 2023. paper

    Takashi Matsubara and Takaharu Yaguchi.

14. Adaptive importance sampling for Deep Ritz. arXiv, 2023. paper

    Xiaoliang Wan, Tao Zhou, and Yuancheng Zhou.

15. DAS-PINNs: A deep adaptive sampling method for solving high-dimensional partial differential equations. JCP, 2023. paper

    Kejun Tang, Xiaoliang Wan, and Chao Yang.

Mesh

1. MeshingNet: A new mesh generation method based on deep learning. ICCS, 2022. paper

   Zheyan Zhang, Yongxing Wang, Peter K. Jimack, and He Wang.

2. M2N: Mesh movement networks for PDE solvers. arXiv, 2022. paper

   Wenbin Song, Mingrui Zhang, Joseph G. Wallwork, Junpeng Gao, Zheng Tian, Fanglei Sun, Matthew D. Piggott, Junqing Chen, Zuoqiang Shi, Xiang Chen, and Jun Wang.

3. RANG: A residual-based adaptive node generation method for physics-informed neural networks. arXiv, 2022. paper

   Wei Peng, Weien Zhou, Xiaoya Zhang, Wen Yao, and Zheliang Liu.

4. Learning a mesh motion technique with application to fluid-structure interaction and shape optimization. arXiv, 2022. paper

   Johannes Haubne and Miroslav Kuchta.

5. Accelerated training of physics-informed neural networks (PINNs) using meshless discretizations. arXiv, 2022. paper

   Ramansh Sharma and Varun Shankar.

6. An improved structured mesh generation method based on physics-informed neural networks. arXiv, 2022. paper

   Xinhai Chen, Jie Liu, Junjun Yan, Zhichao Wang, and Chunye Gong.

7. Mesh-free Eulerian physics-informed neural networks. arXiv, 2022. paper

   Fabricio Arend Torres, Marcello Massimo Negri, Monika Nagy-Huber, Maxim Samarin, and Volker Roth.

8. Fixed-budget online adaptive mesh learning for physics-informed neural networks. Towards parameterized problem inference. arXiv, 2022. paper

   Thi Nguyen Khoa Nguyen, Thibault Dairay, Raphaël Meunier, Christophe Millet, and Mathilde Mougeot.

9. Learning controllable adaptive simulation for multi-resolution physics. ICLR, 2023. paper

   Tailin Wu, Takashi Maruyama, Qingqing Zhao, Gordon Wetzstein, and Jure Leskovec.

10. A closest point method for surface PDEs with interior boundary conditions for geometry processing. arXiv, 2023. paper

    Nathan King, Haozhe Su, Mridul Aanjaneya, Steven Ruuth, and Christopher Batty.

11. Efficient training of physics-informed neural networks with direct grid refinement algorithm. arXiv, 2023. paper

    Shikhar Nilabh and Fidel Grandia.

12. Redefining Super-Resolution: Fine-mesh PDE predictions without classical simulations. arXiv, 2023. paper

    Rajat Kumar Sarkar, Ritam Majumdar, Vishal Jadhav, Sagar Srinivas Sakhinana, and Venkataramana Runkana.

13. MMPDE-Net and moving sampling physics-informed neural networks based on moving mesh method. arXiv, 2023. paper

    Yu Yang, Qihong Yang, Yangtao Deng, and Qiaolin He.

14. Reinforcement learning for adaptive mesh refinement. AISTATS, 2023. paper

    Jiachen Yang, Tarik Dzanic, Brenden Petersen, Jun Kudo, Ketan Mittal, Vladimir Tomov, Jean-Sylvain Camier, Tuo Zhao, Hongyuan Zha, Tzanio Kolev, Robert Anderson, and Daniel Faissol.

Decomposition

1. Composing partial differential equations with physics-aware neural networks. ICLR, 2022. paper

   Matthias Karlbauer, Timothy Praditia, Sebastian Otte, Sergey Oladyshkin, Wolfgang Nowak, and Martin V. Butz.

2. Learning composable energy surrogates for PDE order reduction. NIPS, 2020. paper

   Alex Beatson, Jordan T. Ash, Geoffrey Roeder, Tianju Xue, and Ryan P. Adams.

3. Neural basis functions for accelerating solutions to high mach euler equations. ICML, 2022. paper

   David Witman, Alexander New, Hicham Alkandry, and Honest Mrema.

4. PFNN-2: A domain decomposed penalty-free neural network method for solving partial differential equations. arXiv, 2022. paper

   Hailong Shen and Chao Yang.

5. Finite basis physics-informed neural networks (FBPINNs): A scalable domain decomposition approach for solving differential equations. arXiv, 2021. paper

   Ben Moseley, Andrew Markham, and Tarje Nissen-Meyer.

6. Finite basis physics-informed neural networks as a Schwarz domain decomposition method. arXiv, 2022. paper

   Victorita Dolean, Alexander Heinlein, Siddhartha Mishra, and Ben Moseley.

7. NeuralStagger: Accelerating physics constrained neural PDE solver with spatial-temporal decomposition. ICML, 2023. paper

   Xinquan Huang, Wenlei Shi, Qi Meng, Yue Wang, Xiaotian Gao, Jia Zhang, and Tieyan Liu.

8. Augmenting physical models with deep networksfor complex dynamics forecasting. Journal of Statistical Mechanics: Theory and Experiment, 2021. paper

   Yuan Yin, Vincent Le Guen, Jérémie Dona, Emmanuel de Bézenac, Ibrahim Ayed, Nicolas Thome, and Patrick Gallinari.

9. LordNet: Learning to solve parametric partial differential equations without simulated data. arXiv, 2022. paper

   Wenlei Shi, Xinquan Huang, Xiaotian Gao, Xinran Wei, Jia Zhang, Jiang Bian, Mao Yang, and Tieyan Liu.

10. An optimisation–based domain–decomposition reduced order model for the incompressible Navier-Stokes equations. arXiv, 2022. paper

    Ivan Prusak, Monica Nonino, Davide Torlo, Francesco Ballarin, and Gianluigi Rozza.

11. NUNO: A general gramework for learning parametric PDEs with non-uniform data. ICML, 2023. paper

    Songming Liu, Zhongkai Hao, Chengyang Ying, Hang Su, Ze Cheng, and Jun Zhu.

12. Enhancing training of physics-informed neural networks using domain-decomposition based preconditioning strategies. arXiv, 2023. paper

    Alena Kopaničáková, Hardik Kothari, George Em Karniadakis, and Rolf Krause.

13. Multilevel domain decomposition-based architectures for physics-informed neural networks. arXiv, 2023. paper

    Victorita Dolean, Alexander Heinlein, Siddhartha Mishra, and Ben Moseley.

14. Friedrichs' systems discretized with the Discontinuous Galerkin method: Domain decomposable model order reduction and Graph Neural Networks approximating vanishing viscosity solutions. arXiv, 2023. paper

    Francesco Romor, Davide Torlo, and Gianluigi Rozza.

15. A deep domain decomposition method based on Fourier features. Journal of Computational and Applied Mathematics, 2023. paper

    Sen Li, Yingzhi Xia, Yu Liu, and Qifeng Liao.

16. A unified scalable framework for causal sweeping strategies for physics-informed neural networks (PINNs) and their temporal decompositions. arXiv, 2023. paper

    Michael Penwarden, Ameya D. Jagtap, Shandian Zhe, George Em Karniadakis, and Robert M. Kirby.

17. A generalized Schwarz-type non-overlapping domain decomposition method using physics-constrained neural networks. arXiv, 2023. paper

    Shamsulhaq Basir and Inanc Senocak.

18. Breaking boundaries: Distributed domain decomposition with scalable physics-informed neural PDE solvers. arXiv, 2023. paper

    Arthur Feeney, Zitong Li, Ramin Bostanabad, and Aparna Chandramowlishwaran.

19. Efficient learning of PDEs via Taylor expansion and sparse decomposition into value and Fourier domains. arXiv, 2023. paper

    Md Nasim and Yexiang Xue.

20. Adaptive task decomposition physics-informed neural networks. Computer Methods in Applied Mechanics and Engineering, 2023. paper

    Jianchuan Yang, Xuanqi Liu, Yu Diao, Xi Chen, and Haikuo Hu.

21. PF-DMD: Physics-fusion dynamic mode decomposition for accurate and robust forecasting of dynamical systems with imperfect data and physics. arXiv, 2023. paper

    Yuhui Yin, Chenhui Kou, Shengkun Jia, Lu Lu, Xigang Yuan, and Yiqing Luo.

22. Subspace decomposition based DNN algorithm for elliptic type multi-scale PDEs. JCP, 2023. paper

    Xian Li, Zhiqin John Xu, and Lei Zhang.

23. Adaptive deep Fourier residual method via overlapping domain decomposition. arXiv, 2024. paper

    Jamie M. Taylor, Manuela Bastidas, Victor M. Calo, and David Pardo.

Disentangle

1. PDE-driven spatiotemporal disentanglement. ICLR, 2021. paper

   Jérémie Donà, Jean-Yves Franceschi, Sylvain Lamprier, and Patrick Gallinari.

2. Disentangling physical dynamics from unknown factors for unsupervised video prediction. CVPR, 2020. paper

   Vincent Le Guen and Nicolas Thome.

Solver

1. Solver-in-the-loop: Learning from differentiable physics to interact with iterative PDE-solvers. NIPS, 2020. paper

   Kiwon Um, Robert Brand, Yun (Raymond) Fei, Philipp Holl, and Nils Thuerey.

2. Lie point symmetry data augmentation for neural PDE solvers. ICML, 2022. paper

   Johannes Brandstetter, Max Welling, and Daniel E. Worrall.

3. Incorporating symmetry into deep dynamics models for improved generalization. ICLR, 2021. paper

   Rui Wang, Robin Walters, and Rose Yu.

4. HyperSolvers: Toward fast continuous-depth models. NIPS, 2020. paper

   Michael Poli, Stefano Massaroli, Atsushi Yamashita, Hajime Asama, and Jinkyoo Park.

5. PIXEL: Physics-informed cell representations for fast and accurate PDE solvers. NIPS, 2022. paper

   Namgyu Kang, Byeonghyeon Lee, Youngjoon Hong, Seok-Bae Yun, and Eunbyung Park.

6. NeuralSim: Augmenting differentiable simulators with neural networks. ICRA, 2021. paper

   Eric Heiden, David Millard, Erwin Coumans, Yizhou Sheng, and Gaurav S. Sukhatme.

7. DPM: A deep learning PDE augmentation method with application to large-eddy simulation. JCP, 2020. paper

   Justin Sirignano, Jonathan F.MacArt, and Jonathan B.Freund.

8. General covariance data augmentation for neural PDE solvers. ICML, 2023. paper

   Vladimir Fanaskov, Tianchi Yu, Alexander Rudikov, and Ivan Oseledets.

9. Learning preconditioners for conjugate gradient PDE solvers. ICML, 2023. paper

   Yichen Li, Peter Yichen Chen, Tao Du, and Wojciech Matusik.

10. Autoregressive Renaissance in neural PDE solvers. ICLR, 2023. paper

    Yolanne Yi Ran Lee.

11. Stability of implicit neural networks for long-term forecasting in dynamical systems. ICLR, 2023. paper

    Leon Migus, Julien Salomon, and Patrick Gallinari.

12. Training deep surrogate models with large scale online learning. ICML, 2023. paper

    Lucas Meyer, Marc Schouler, Robert A. Caulk, Alejandro Ribes, and Bruno Raffin.

13. Self-supervised learning with Lie symmetries for partial differential equations. arXiv, 2023. paper

    Grégoire Mialon, Quentin Garrido, Hannah Lawrence, Danyal Rehman, Yann LeCun, and Bobak T. Kiani.

14. Interpretable neural PDE solvers using symbolic frameworks. arXiv, 2023. paper

    Yolanne Yi Ran Lee.

15. Modeling accurate long rollouts with temporal neural PDE solvers. ICML, 2023. paper

    Phillip Lippe, Bastiaan S. Veeling, Paris Perdikaris, Richard E. Turner, and Johannes Brandstetter.

16. Neural-integrated meshfree (NIM) method: A differentiable programming-based hybrid solver for computational mechanics. arXiv, 2023. paper

    Honghui Du and Qizhi He.

17. A deep-genetic algorithm (deep-GA) approach for high-dimensional nonlinear parabolic partial differential equations. Computers & Mathematics with Applications, 2023. paper

    Endah R.M. Putri, Muhammad L. Shahab, Mohammad Iqbal, Imam Mukhlash, Amirul Hakam, Lutfi Mardianto, and Hadi Susanto.

18. Physics-enhanced deep surrogates for partial differential equations. NMI, 2023. paper

    Raphaël Pestourie, Youssef Mroueh, Chris Rackauckas, Payel Das, and Steven G. Johnson .

19. Better neural PDE solvers through data-free mesh movers. arXiv, 2023. paper

    Peiyan Hu, Yue Wang, and Zhiming Ma.

20. Generating synthetic data for neural operators. arXiv, 2024. paper

    Erisa Hasani and Rachel A. Ward.

21. A deep branching solver for fully nonlinear partial differential equations. JCP, 2024. paper

    Jiang Yu Nguwi, Guillaume Penent, and Nicolas Privault.

22. Accelerating data generation for neural operators via Krylov subspace recycling. ICLR, 2024. paper

    Hong Wang, Zhongkai Hao, Jie Wang, Zijie Geng, Zhen Wang, Bin Li, and Feng Wu.

23. Space and time continuous physics simulation from partial observations. ICLR, 2024. paper

    Janny Steeven, Nadri Madiha, Digne Julie, and Wolf Christian.

AutoML

1. Auto-PINN: Understanding and optimizing physics-informed neural architecture. arXiv, 2022. paper

   Yicheng Wang, Xiaotian Han, Chiayuan Chang, Daochen Zha, Ulisses Braga-Neto, and Xia Hu.

2. Building high accuracy emulators for scientific simulations with deep neural architecture search. Machine Learning: Science and Technology, 2022. paper

   M F Kasim, D Watson-Parris, L Deaconu, S Oliver, P Hatfield, D H Froula, G Gregori, M Jarvis, S Khatiwala, J Korenaga, J Topp-Mugglestone, E Viezzer, and S M Vinko.

3. Learning operations for neural PDE solvers. ICLR, 2019. paper

   Nicholas Roberts, Mikhail Khodak, Tri Dao, Liam Li, Christopher Re, and Ameet Talwalkar.

4. AutoPINN: When AutoML meets physics-informed neural networks. arXiv, 2022. paper

   Xinle Wu, Dalin Zhang, Miao Zhang, Chenjuan Guo, Shuai Zhao, Yi Zhang, Huai Wang, and Bin Yang.

5. NAS-PINN: Neural architecture search-guided physics-informed neural network for solving PDEs. arXiv, 2023. paper

   Yifan Wang and Linlin Zhong.

Neural Implicit Flow

1. Neural implicit flow: A mesh-agnostic dimensionality reduction paradigm of spatio-temporal data. arXiv, 2022. paper

   Shaowu Pan, Steven L. Brunton, and J. Nathan Kutz.

2. CROM: Continuous reduced-order modeling of PDEs using implicit neural representations. ICLR, 2023. paper

   Peter Yichen Chen, Jinxu Xiang, Dong Heon Cho, Yue Chang, G A Pershing, Henrique Teles Maia, Maurizio Chiaramonte, Kevin Carlberg, and Eitan Grinspun.

3. MAgNet: Mesh agnostic neural PDE solver. NIPS, 2022. paper

   Oussama Boussif, Dan Assouline, Loubna Benabbou, and Yoshua Bengio.

4. NTopo: Mesh-free topology optimization using implicit neural representations. NIPS, 2021. paper

   Jonas Zehnder, Yue Li, Stelian Coros, and Bernhard Thomaszewski.

5. ContactNets: Learning discontinuous contact dynamics with smooth, implicit representations. ICLR, 2020. paper

   Samuel Pfrommer, Mathew Halm, and Michael Posa.

6. Continuous PDE dynamics forecasting with implicit neural representations. ICLR, 2023. paper

   Yuan Yin, Matthieu Kirchmeyer, Jean-Yves Franceschi, Alain Rakotomamonjy, and Patrick Gallinari.

7. Stability of implicit neural networks for long-term forecasting in dynamical systems. ICLR, 2023. paper

   Leon Migus, Julien Salomon, and Patrick Gallinari.

8. Operator learning with neural fields: Tackling PDEs on general geometries. arXiv, 2023. paper

   Louis Serrano, Lise Le Boudec, Armand Kassaï Koupaï, Thomas X Wang, Yuan Yin, Jean-Noël Vittaut, and Patrick Gallinari.

9. Accelerated solutions of convection-dominated partial differential equations using implicit feature tracking and empirical quadrature. arXiv, 2023. paper

   Marzieh Alireza Mirhoseini and Matthew J. Zahr.

10. Implicit neural spatial representations for time-dependent PDEs. ICML, 2023. paper

    Honglin Chen, Rundi Wu, Eitan Grinspun, Changxi Zheng, and Peter Yichen Chen.

11. Reduced-order modeling for parameterized PDEs via implicit neural representations. NIPS, 2023. paper

    Tianshu Wen, Kookjin Lee, and Youngsoo Choi.

Uncertainty Quantification

1. Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems. arXiv, 2021. paper

   Dongkun Zhang, Lu Lu, Ling Guo, and George Em Karniadakis.

2. Adversarial uncertainty quantification in physics-informed neural networks. JCP, 2021. paper

   Yibo Yang and Paris Perdikaris.

3. Conditional Karhunen-Loève expansion for uncertainty quantification and active learning in partial differential equation models. JCP, 2020. paper

   Ramakrishna Tipireddy, David A.Barajas-Solano, and Alexandre M.Tartakovsky.

4. Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data. JCP, 2019. paper

   Yinhao Zhu, Nicholas Zabarasa, Phaedon-Stelios Koutsourelakisb, and Paris Perdikaris.

5. Error-aware B-PINNs: Improving uncertainty quantification in Bayesian physics-informed neural networks. arXiv, 2022. paper

   Olga Graf, Pablo Flores, Pavlos Protopapas, and Karim Pichara.

6. Physics-informed information field theory for modeling physical systems with uncertainty quantification. arXiv, 2023. paper

   Alex Alberts and Ilias Bilionis.

7. Quantifying uncertainty for deep learning based forecasting and flow-reconstruction using neural architecture search ensembles. arXiv, 2023. paper

   Romit Maulik, Romain Egele, Krishnan Raghavan, and Prasanna Balaprakash.

8. Physics-informed variational inference for uncertainty quantification of stochastic differential equations. JCP, 2023. paper

   Hyomin Shin and Minseok Choi.

9. Uncertainty quantification in scientific machine learning: Methods, metrics, and comparisons. JCP, 2023. paper

   Apostolos F. Psaros, Xuhui Meng, Zongren Zou, Ling Guo, and George Em Karniadakis.

10. Auto-weighted Bayesian physics-informed neural networks and robust estimations for multitask inverse problems in pore-scale imaging of dissolution. arXiv, 2023. paper

    Sarah Perez and Philippe Poncet.

11. Neural SPDE solver for uncertainty quantification in high-dimensional space-time dynamics. arXiv, 2023. paper

    Maxime Beauchamp, Hugo Georgenthum, and Ronan Fablet.

12. Ensemble models outperform single model uncertainties and predictions for operator-learning of hypersonic flows. NIPS, 2023. paper

    Victor J. Leon, Noah Ford, Honest Mrema, Jeffrey Gilbert, and Alexander New.

13. Evaluating Uncertainty Quantification approaches for Neural PDEs in scientific applications. NIPS, 2023. paper

    Vardhan Dongre and Gurpreet Singh Hora.

14. Uncertainty quantification for noisy inputs-outputs in physics-informed neural networks and neural operators. arXiv, 2023. paper

    Zongren Zou, Xuhui Meng, and George Em Karniadakis.

15. Data-driven autoencoder numerical solver with uncertainty quantification for fast physical simulations. arXiv, 2023. paper

    Christophe Bonneville, Youngsoo Choi, Debojyoti Ghosh, and Jonathan L. Belof.

16. Randomized physics-informed machine learning for uncertainty quantification in high-dimensional inverse problems. arXiv, 2023. paper

    Yifei Zong, David Barajas-Solano, and Alexandre M. Tartakovsky.

Generative Model

1. A framework for data-driven solution and parameter estimation of PDEs using conditional generative adversarial networks. NCS, 2021. paper

   Teeratorn Kadeethum, Daniel O’Malley, Jan Niklas Fuhg, Youngsoo Choi, Jonghyun Lee, Hari S. Viswanathan, and Nikolaos Bouklas.

2. Neural SDEs as infinite-dimensional GANs. ICML, 2021. paper

   Patrick Kidger, James Foster, Xuechen Li, and Terry J Lyons.

3. PID-GAN: A GAN framework based on a physics-informed discriminator for uncertainty quantification with physics. KDD, 2021. paper

   Arka Daw, M. Maruf, and Anuj Karpatne.

4. Generative adversarial neural operators. Transactions on Machine Learning Research, 2022. paper

   Md Ashiqur Rahman, Manuel A Florez, Anima Anandkumar, Zachary E Ross, and Kamyar Azizzadenesheli.

5. Weak adversarial networks for high-dimensional partial differential equations. JCP, 2020. paper

   Yaohua Zang, Gang Bao, Xiaojing Ye, and Haomin Zhou.

6. Wasserstein generative adversarial uncertainty quantification in physics-informed neural networks. JCP, 2022. paper

   Yihang Gao and Michael K. Ng.

7. Competitive physics informed networks. ICLR, 2023. paper

   Qi Zeng, Yash Kothari, Spencer H Bryngelson, and Florian Tobias Schaefer.

8. Learning generative neural networks with physics knowledge. Research in the Mathematical Sciences, 2022. paper

   Kailai Xu, Weiqiang Zhu, and Eric Darve.

9. Diffusion generative models in infinite dimensions. arXiv, 2022. paper

   Gavin Kerrigan, Justin Ley, and Padhraic Smyth.

10. Revisiting PINNs: Generative adversarial physics-informed neural networks and point-weighting method. arXiv, 2022. paper

    Wensheng Li, Chao Zhang, Chuncheng Wang, Hanting Guan, and Dacheng Tao.

11. PIAT: Physics informed adversarial training for solving partial differential equations. arXiv, 2022. paper

    Simin Shekarpaz, Mohammad Azizmalayeri, and Mohammad Hossein Rohban.

12. Physics-constrained generative adversarial networks for 3D turbulence. arXiv, 2022. paper

    Dima Tretiak, Arvind T. Mohan, and Daniel Livescu.

13. Score-based diffusion models in function space. arXiv, 2023. paper

    Jae Hyun Lim, Nikola B. Kovachki, Ricardo Baptista, Christopher Beckham, Kamyar Azizzadenesheli, Jean Kossaifi, Vikram Voleti, Jiaming Song, Karsten Kreis, Jan Kautz, Christopher Pal, Arash Vahdat, and Anima Anandkumar.

14. Infinite-dimensional diffusion models for function spaces. arXiv, 2023. paper

    Jakiw Pidstrigach, Youssef Marzouk, Sebastian Reich, and Sven Wang.

15. A physics-informed diffusion model for high-fidelity flow field reconstruction. JCP, 2023. paper

    Dule Shu, Zijie Li, and Amir Barati Farimani.

16. Generative modelling with inverse heat dissipation. ICLR, 2023. paper

    Severi Rissanen, Markus Heinonen, and Arno Solin.

17. Differentiable Gaussianization layers for inverse problems regularized by deep generative models. ICLR, 2023. paper

    Dongzhuo Li.

18. Transformer meets boundary value inverse problems. ICLR, 2023. paper

    Ruchi Guo, Shuhao Cao, and Long Chen.

19. ViTO: Vision Transformer-operator. arXiv, 2023. paper

    Oded Ovadia, Adar Kahana, Panos Stinis, Eli Turkel, and George Em Karniadakis.

20. LatentPINNs: Generative physics-informed neural networks via a latent representation learning. arXiv, 2023. paper

    Mohammad H. Taufik and Tariq Alkhalifah.

21. Generative diffusion learning for parametric partial differential equations. arXiv, 2023. paper

    Ting Wang, Petr Plechac, and Jaroslaw Knap.

22. Scalable Transformer for PDE surrogate modeling. arXiv, 2023. paper

    Zijie Li, Dule Shu, and Amir Barati Farimani.

23. Latent traversals in generative models as potential flows. ICML, 2023. paper

    Yue Song, T. Anderson Keller, Nicu Sebe, and Max Welling.

24. Learning space-time continuous neural PDEs from partially observed states. arXiv, 2023. paper

    Valerii Iakovlev, Markus Heinonen, and Harri Lähdesmäki.

25. DiTTO: Diffusion-inspired temporal Transformer operator. arXiv, 2023. paper

    Oded Ovadia, Eli Turkel, Adar Kahana, and George Em Karniadakis.

26. PINNsFormer: A Transformer-based framework For physics-informed neural networks. arXiv, 2023. paper

    Leo Zhiyuan Zhao, Xueying Ding, and B. Aditya Prakash.

27. CoCoGen: Physically-consistent and conditioned score-based generative models for forward and inverse problems. arXiv, 2023. paper

    Christian Jacobsen, Yilin Zhuang, and Karthik Duraisamy.

Transformer

1. Learning operators with coupled attention. JMLR, 2022. paper

   Matthew Thorpe, Tan Minh Nguyen, Hedi Xia, Thomas Strohmer, Andrea Bertozzi, Stanley Osher, and Bao Wang.

2. Choose a Transformer: Fourier or Galerkin. NIPS, 2021. paper

   Shuhao Cao.

3. Predicting physics in mesh-reduced space with temporal attention. ICLR, 2022. paper

   Xu Han, Han Gao, Tobias Pfaff, Jianxun Wang, and Liping Liu.

4. SiT: Simulation transformer for particle-based physics simulation. ICLR, 2022. paper

   Yidi Shao, Chen Change Loy, and Bo Dai.

5. Physics-informed long-sequence forecasting from multi-resolution spatiotemporal data. IJCAI, 2022. paper

   Chuizheng Meng, Hao Niu, Guillaume Habault, Roberto Legaspi, Shinya Wada, Chihiro Ono, and Yan Liu.

6. TransFlowNet: A physics-constrained Transformer framework for spatio-temporal super-resolution of flow simulations. Journal of Computational Science, 2022. paper

   Xinjie Wang, Siyuan Zhu, Yundong Guo, Peng Han, Yucheng Wang, Zhiqiang Wei, and Xiaogang Jin.

7. Self-adaptive physics-informed neural networks using a soft attention mechanism. AAAI-MLPS, 2021. paper

   Levi D. McClenny and Ulisses Braga-Neto.

8. Transformer for partial differential equations' operator learning. arXiv, 2022. paper

   Zijie Li, Kazem Meidani, and Amir Barati Farimani.

9. Physics-informed attention-based neural network for hyperbolic partial differential equations: Application to the Buckley–Leverett problem. Scientific Reports, 2022. paper

   Ruben Rodriguez-Torrado, Pablo Ruiz, Luis Cueto-Felgueroso, Michael Cerny Green, Tyler Friesen, Sebastien Matringe, and Julian Togelius.

10. Self-adaptive physics-informed neural networks using a soft attention mechanism. AAAI-MLPS, 2021. paper

    Levi Mc Clenny and Ulisses Braga-Neto.

11. Nonlinear reconstruction for operator learning of PDEs with discontinuities. ICLR, 2023. paper

    Samuel Lanthaler, Roberto Molinaro, Patrik Hadorn, and Siddhartha Mishra.

12. GNOT: A general neural operator transformer for operator learning. ICML, 2023. paper

    Zhongkai Hao, Chengyang Ying, Zhengyi Wang, Hang Su, Yinpeng Dong, Songming Liu, Ze Cheng, Jun Zhu, and Jian Song.

13. In-context operator learning for differential equation problems. arXiv, 2023. paper

    Liu Yang, Siting Liu, Tingwei Meng, and Stanley J. Osher.

14. Learning neural PDE solvers with parameter-guided channel attention. ICML, 2023. paper

    Makoto Takamoto, Francesco Alesiani, and Mathias Niepert.

15. Physics informed token Transformer. arXiv, 2023. paper

    Cooper Lorsung, Zijie Li, and Amir Barati Farimani.

16. Improved operator learning by orthogonal attention. arXiv, 2023. paper

    Zipeng Xiao, Zhongkai Hao, Bokai Lin, Zhijie Deng, and Hang Su.

17. Multi-scale time-stepping of partial differential equations with Transformers. arXiv, 2023. paper

    AmirPouya Hemmasian and Amir Barati Farimani.

18. Deciphering and integrating invariants for neural operator learning with various physical mechanisms. arXiv, 2023. paper

    Rui Zhang, Qi Meng, and Zhiming Ma.

19. Attention-enhanced neural differential equations for physics-informed deep learning of ion transport. NIPS, 2023. paper

    Danyal Rehman and John H. Lienhard.

20. Inducing point operator Transformer: A flexible and scalable architecture for solving PDEs. arXiv, 2023. paper

    Seungjun Lee and Taeil Oh.

21. Loss-attentional physics-informed neural networks. JCP, 2024. paper

    Yanjie Song, He Wang, He Yang, Maria Luisa Taccari, and Xiaohui Chen.

Theory

1. Convergence analysis of a quasi-Monte Carlo-based deep learning algorithm for solving partial differential equations. arXiv, 2022. paper

   Fengjiang Fu and Xiaoqun Wang.

2. Solving PDEs by variational physics-informed neural networks: A posteriori error analysis. arXiv, 2022. paper

   Stefano Berrone, Claudio Canuto, and Moreno Pintore.

3. A unified framework for the error analysis of physics-informed neural networks. arXiv, 2023. paper

   Marius Zeinhofer, Rami Masri, and Kent-André Mardal.

4. Neural tangent kernel analysis of PINN for advection-diffusion equation. arXiv, 2022. paper

   M. H. Saadat, B. Gjorgiev, L. Das, and G. Sansavini.

5. Error estimates for DeepONets: A deep learning framework in infinite dimensions. Transactions of Mathematics and Its Applications, 2022. paper

   Samuel Lanthaler, Siddhartha Mishra, and George Em Karniadakis.

6. Hutchinson trace estimation for high-dimensional and high-order physics-informed neural networks. arXiv, 2023. paper

   Zheyuan Hu, Zekun Shi, George Em Karniadakis, and Kenji Kawaguchi.

7. An analysis of universal differential equations for data-driven discovery of ordinary differential equations. arXiv, 2023. paper

   Mattia Silvestri, Federico Baldo, Eleonora Misino, and Michele Lombardi.

8. Exponential convergence of deep operator networks for elliptic partial differential equations. SIAM Journal on Numerical Analysis, 2023. paper

   Carlo Marcati and Christoph Schwab.

9. A discretization-invariant extension and analysis of some deep operator networks. arXiv, 2023. paper

   Zecheng Zhang, Wing Tat Leung, and Hayden Schaeffer.

10. Machine learning for elliptic PDEs: Fast rate generalization bound, neural scaling law and minimax optimality. ICLR, 2022. paper

    Yiping Lu, Haoxuan Chen, Jianfeng Lu, Lexing Ying, and Jose Blanchet.

11. Elliptic PDE learning is provably data-efficient. arXiv, 2023. paper

    Nicolas Boullé, Diana Halikias, and Alex Townsend.

12. Error estimates of residual minimization using neural networks for linear PDEs. Journal of Machine Learning for Modeling and Computing, 2023. paper

    Yeonjong Shin, Zhongqiang Zhang, and George Em Karniadakis.

13. Analysis of the generalization error of deep learning based on randomized quasi-Monte Carlo for solving linear Kolmogorov PDEs. arXiv, 2023. paper

    Jichang Xiao, Fengjiang Fu, and Xiaoqun Wang.

14. Rigorous a posteriori error bounds for PDE-defined PINNs. TNNLS, 2023. paper

    Birgit Hillebrecht and Benjamin Unger.

15. Error estimation for physics-informed neural networks with implicit Runge-Kutta methods. arXiv, 2023. paper

    Jochen Stiasny and Spyros Chatzivasileiadis.

16. Approximation of solution operators for high-dimensional PDEs. arXiv, 2024. paper

    Nathan Gaby and Xiaojing Ye.

17. Accuracy analysis of physics-informed neural networks for approximating the critical SQG equation. arXiv, 2024. paper

    Elie Abdo, Ruimeng Hu, and Quyuan Lin.

18. Deep Ritz method for elliptical multiple eigenvalue problems. Journal of Scientific Computing, 2024. paper

    Xia Ji, Yuling Jiao, Xiliang Lu, Pengcheng Song, and Fengru Wang.

19. Inf-Sup neural networks for high-dimensional elliptic PDE problems. arXiv, 2024. paper

    Xiaokai Huo and Hailiang Liu.

Gaussian Process

1. PAGP: A physics-assisted Gaussian process framework with active learning for forward and inverse problems of partial differential equations. arXiv, 2022. paper

   Jiahao Zhang, Shiqi Zhang, and Guang Lin.

2. Solving and learning nonlinear PDEs with Gaussian processes. JCP, 2021. paper

   Yifan Chen, Bamdad Hosseini, Houman Owhadi, and Andrew M.Stuart.

3. Neural-net-induced Gaussian process regression for function approximation and PDE solution. JCP, 2019. paper

   Guofei Pang, Liu Yang, and George Em Karniadakis.

4. Learning neural optimal interpolation models and solvers. arXiv, 2022. paper

   Maxime Beauchamp, Joseph Thompson, Hugo Georgenthum, Quentin Febvre, and Ronan Fablet.

5. Inference of nonlinear partial differential equations via constrained Gaussian processes. arXiv, 2022. paper

   Zhaohui Li, Shihao Yang, and Jeff Wu.

6. Gaussian process priors for systems of linear partial differential equations with constant coefficients. arXiv, 2022. paper

   Marc Härkönen, Markus Lange-Hegermann, and Bogdan Raiţă.

7. Sparse Cholesky factorization for solving nonlinear PDEs via Gaussian processes. arXiv, 2023. paper

   Yifan Chen, Houman Owhadi, and Florian Schäfer.

8. A mini-batch method for solving nonlinear PDEs with Gaussian processes. arXiv, 2023. paper

   Xianjin Yang and Houman Owhadi.

9. Random grid neural processes for parametric partial differential equations. ICML, 2023. paper

   Arnaud Vadeboncoeur, Ieva Kazlauskaite, Yanni Papandreou, Fehmi Cirak, Mark Girolami, and Omer Deniz Akyildiz.

10. Gaussian process priors for systems of linear partial differential equations with constant coefficients. ICML, 2023. paper

    Marc Harkonen, Markus Lange-Hegermann, and Bogdan Raita.

11. GPLaSDI: Gaussian process-based interpretable latent space dynamics identification through deep autoencoder. arXiv, 2023. paper

    Christophe Bonneville, Youngsoo Choi, Debojyoti Ghosh, and Jonathan L.Belof.

12. Solving high frequency and multi-scale PDEs with Gaussian processes. ICLR, 2024. paper

    Shikai Fang, Madison Cooley, Da Long, Shibo Li, Robert Kirby, and Shandian Zhe.

13. Physics-constrained convolutional neural networks for inverse problems in spatiotemporal partial differential equations. arXiv, 2024. paper

    Daniel Kelshaw and Luca Magri.

Variation

1. PI-VAE: Physics-informed variational auto-encoder for stochastic differential equations. Computer Methods in Applied Mechanics and Engineering, 2022. paper

   Weiheng Zhong and Hadi Meidani.

2. Robust SDE-based variational formulations for solving linear PDEs via deep learning. ICML, 2022. paper

   Lorenz Richter and Julius Berner.

3. HP-VPINNs: Variational physics-informed neural networks with domain decomposition. Computer Methods in Applied Mechanics and Engineering, 2021. paper

   Ehsan Kharazmi, Zhongqiang Zhang, and George Em Karniadakis.

4. Variational onsager neural networks (VONNs): A thermodynamics-based variational learning strategy for non-equilibrium PDEs. Journal of the Mechanics and Physics of Solids, 2022. paper

   Shenglin Huang, Zequn He, Bryan Chem, and Celia Reina.

5. Variational Monte Carlo approach to partial differential equations with neural networks. arXiv, 2022. paper

   Moritz Reh and Martin Gärttner.

6. Energetic variational neural network discretizations to gradient flows. arXiv, 2022. paper

   Ziqing Hu, Chun Liu, Yiwei Wang, and Zhiliang Xu.

7. Variational Bayes deep operator network: A data-driven Bayesian solver for parametric differential equations. arXiv, 2022. paper

   Shailesh Garg and Souvik Chakraborty.

8. Variational inference in neural functional prior using normalizing flows: Application to differential equation and operator learning problems. arXiv, 2023. paper

   Xuhui Meng.

9. Neural network approximations of PDEs beyond linearity: A representational perspective. ICML, 2023. paper

   Tanya Marwah, Zachary Chase Lipton, Jianfeng Lu, and Andrej Risteski.

Bayesian

1. Bayesian deep learning for partial differential equation parameter discovery with sparse and noisy data. JCP: X, 2022. paper

   Christophe Bonneville and Christopher Earls.

2. B-PINNs: Bayesian physics-informed neural networks for forward and inverse PDE problems with noisy data. JCP, 2021. paper

   Liu Yang, Xuhui Meng, and George Em Karniadakis.

3. Approximate Bayesian neural operators: Uncertainty quantification for parametric PDEs. arXiv, 2022. paper

   Emilia Magnani, Nicholas Krämer, Runa Eschenhagen, Lorenzo Rosasco, and Philipp Hennig.

4. Bayesian autoencoders for data-driven discovery of coordinates, governing equations and fundamental constants. arXiv, 2022. paper

   L. Mars Gao and J. Nathan Kutz.

5. Bayesian physics informed neural networks for data assimilation and spatio-temporal modelling of wildfires. arXiv, 2022. paper

   Joel Janek Dabrowski, Daniel Edward Pagendam, James Hilton, Conrad Sanderson, Daniel MacKinlay, Carolyn Huston, Andrew Bolt, and Petra Kuhnert.

6. Bayesian deep operator learning for homogenized to fine-scale maps for multiscale PDE. arXiv, 2023. paper

   Zecheng Zhang, Christian Moya, Wing Tat Leung, Guang Lin, and Hayden Schaeffer.

Latent Space

1. Multiscale simulations of complex systems by learning their effective dynamics. NMI, 2022. paper

   Pantelis R. Vlachas, Georgios Arampatzis, Caroline Uhler, and Petros Koumoutsakos.

2. A latent space solver for PDE generalization. ICLR, 2021. paper

   Rishikesh Ranade, Chris Hill, Haiyang He, Amir Maleki, and Jay Pathak.

3. Approximate latent force model inference. AAAI, 2021. paper

   Jacob D. Moss, Felix L. Opolka, Bianca Dumitrascu, and Pietro Lió.

4. Learning to accelerate partial differential equations via latent global evolution. NIPS, 2022. paper

   Tailin Wu, Takashi Maruyama, and Jure Leskovec.

5. Exploring physical latent spaces for deep learning. arXiv, 2022. paper

   Chloe Paliard, Nils Thuerey, and Kiwon Um.

6. Certified data-driven physics-informed greedy auto-encoder simulator. arXiv, 2022. paper

   Xiaolong He, Youngsoo Choi, William D. Fries, Jonathan L. Belof, and Jiun-Shyan Chen.

7. Deep latent regularity network for modeling stochastic partial differential equations. AAAI, 2023. paper

   Shiqi Gong, Peiyan Hu, Qi Meng, Yue Wang, Rongchan Zhu, Bingguang Chen, Zhiming Ma, Hao Ni, and Tieyan Liu.

8. Learning in latent spaces improves the predictive accuracy of deep neural operators. arXiv, 2023. paper

   Katiana Kontolati, Somdatta Goswami, George Em Karniadakis, and Michael D. Shields.

9. Solving high-dimensional PDEs with latent spectral models. ICML, 2023. paper

   Haixu Wu, Tengge Hu, Huakun Luo, Jianmin Wang, and Mingsheng Long.

10. Physics-informed generator-encoder adversarial networks with latent space matching for stochastic differential equations. arXiv, 2023. paper

    Ruisong Gao, Min Yang, and Jin Zhang.

Lagrangian

1. Lagrangian PINNs: A causality–conforming solution to failure modes of physics-informed neural networks. arXiv, 2022. paper

   Rambod Mojgani, Maciej Balajewicz, and Pedram Hassanzadeh.

2. AL-PINNs: Augmented Lagrangian relaxation method for physics-informed neural networks. arXiv, 2022. paper

   Hwijae Son, Sung Woong Cho, and Hyung Ju Hwang.

3. Lagrangian flow networks for conservation laws. arXiv, 2023. paper

   Fabricio Arend Torres, Marcello Massimo Negri, Marco Inversi, and Jonathan Aellen.

4. An adaptive augmented Lagrangian method for training physics and equality constrained artificial neural networks. arXiv, 2023. paper

   Shamsulhaq Basir and Inanc Senocak.

5. Constrained optimization via exact augmented Lagrangian and randomized iterative sketching. ICML, 2023. paper

   Ilgee Hong, Sen Na, Michael W. Mahoney, and Mladen Kolar.

6. An adaptive augmented Lagrangian method for training physics and equality constrained artificial neural networks. arXiv, 2023. paper

   Shamsulhaq Basir and Inanc Senocak.

7. Partitioned neural network approximation for partial differential equations enhanced with Lagrange multipliers and localized loss functions. arXiv, 2023. paper

   Deok-Kyu Jang, Kyungsoo Kim, and Hyea Hyun Kim.

Multi Scale

1. Hierarchical deep learning of multiscale differential equation time-steppers. Philosophical Transactions of the Royal Society A, 2022. paper

   Yuying Liu, J. Nathan Kutz, and Steven L. Brunton.

2. NH-PINN: Neural homogenization-based physics-informed neural network for multiscale problems. JCP, 2022. paper

   Wing Tat Leung, Guang Lin, and Zecheng Zhang.

3. Deep multiscale model learning. JCP, 2020. paper

   Yating Wang, Siu Wun Cheung, Eric T.Chung, Yalchin Efendiev, and Min Wang.

4. Multi-scale deep neural networks for solving high dimensional PDEs. arXiv, 2019. paper

   Samuel H. Rudy, Steven L. Brunton, Joshua L. Proctor, and J. Nathan Kutz.

5. Towards multi-spatiotemporal-scale generalized PDE modeling. arXiv, 2022. paper

   Jayesh K. Gupta and Johannes Brandstetter.

6. MultiAdam: Parameter-wise scale-invariant optimizer for multiscale training of physics-informed neural networks. ICML, 2023. paper

   Jiachen Yao, Chang Su, Zhongkai Hao, Songming Liu, Hang Su, and Jun Zhu.

7. Learning homogenization for elliptic operators. arXiv, 2023. paper

   Kaushik Bhattacharya, Nikola Kovachki, Aakila Rajan, Andrew M. Stuart, and Margaret Trautner.

8. Multi-grade deep learning for partial differential equations with applications to the Burgers equation. arXiv, 2023. paper

   Yuesheng Xu and Taishan Zeng.

9. Multiscale neural operators for solving time-independent PDEs. NIPS, 2023. paper

   Winfried Ripken, Lisa Coiffard, Felix Pieper, and Sebastian Dziadzio.

10. Multilevel scalable solvers for stochastic linear and nonlinear problems. arXiv, 2023. paper

    Sudhi Sharma, Pierre Jolivet, Victorita Dolean, and Abhijit Sarkar.

11. Local convolution enhanced global Fourier neural operator for multiscale dynamic spaces prediction. arXiv, 2023. paper

    Xuanle Zhao, Yue Sun, Tielin Zhang, and Bo Xu.

Multi Fidelity

1. Multifidelity deep operator networks. arXiv, 2022. paper

   Amanda A. Howard, Mauro Perego, George Em Karniadakis, and Panos Stinis.

2. Physics and equality constrained artificial neural networks: Application to forward and inverse problems with multi-fidelity data fusion. JCP, 2022. paper

   Lulu Zhang, Tao Luo, Yaoyu Zhang, Weinan E, Zhiqin John Xu, and Zheng Ma.

3. A composite neural network that learns from multi-fidelity data: Application to function approximation and inverse PDE problems. JCP, 2020. paper

   Xuhui Meng and George Em Karniadakis.

4. Multifidelity deep neural operators for efficient learning of partial differential equations with application to fast inverse design of nanoscale heat transport. Physical Review Research, 2022. paper

   Lu Lu, Raphaël Pestourie, Steven G. Johnson, and Giuseppe Romano.

5. Multi-fidelity reduced-order surrogate modeling. arXiv, 2023. paper

   Paolo Conti, Mengwu Guo, Andrea Manzoni, Attilio Frangi, Steven L. Brunton, and J. Nathan Kutz.

6. Multifidelity deep operator networks for data-driven and physics-informed problems. JCP, 2023. paper

   Amanda A. Howard, Mauro Perego, George Em Karniadakis, and Panos Stinis.

7. A multi-fidelity machine learning based semi-Lagrangian finite volume scheme for linear transport equations and the nonlinear Vlasov-Poisson system. arXiv, 2023. paper

   Yongsheng Chen, Wei Guo, and Xinghui Zhong.

8. Neural operator-based super-fidelity: A warm-start approach for accelerating steady-state simulations. arXiv, 2023. paper

   Xuhui Zhou, Jiequn Han, Muhammad I. Zafar, Christopher J. Roy, Heng Xiao.

9. Multi-resolution partial differential equations preserved learning framework for spatiotemporal dynamics. Communications Physics, 2024. paper

   Xinyang Liu, Min Zhu, Lu Lu, Hao Sun, Jianxun Wang.

Multi Grid

1. Learning to optimize multigrid PDE solvers. ICML, 2019. paper

   Daniel Greenfeld, Meirav Galun, Ronen Basri, Irad Yavneh, and Ron Kimmel.

2. Reducing operator complexity in algebraic multigrid with machine learning approaches. arXiv, 2023. paper

   Ru Huang, Kai Chang, Huan He, Ruipeng Li, and Yuanzhe Xi.

3. Multi-grid tensorized Fourier neural operator for high-resolution PDEs. arXiv, 2023. paper

   Jean Kossaifi, Nikola Kovachki, Kamyar Azizzadenesheli, and Anima Anandkumar.

4. MgNO: Efficient parameterization of linear operators via multigrid. arXiv, 2023. paper

   Juncai He, Xinliang Liu, and Jinchao Xu.

5. MGCNN: A learnable multigrid solver for linear PDEs on structured grids. arXiv, 2023. paper

   Yan Xie, Minrui Lv, and Chensong Zhang.

Active Learning

1. Neural Galerkin scheme with active learning for high-dimensional evolution equations. arXiv, 2022. paper

   Joan Bruna, Benjamin Peherstorfer, and Eric Vanden-Eijnden.

2. Discovering and forecasting extreme events via active learning in neural operators. arXiv, 2022. paper

   Ethan Pickering, Stephen Guth, George Em Karniadakis, and Themistoklis P. Sapsis.

3. Active learning based sampling for high-dimensional nonlinear partial differential equations. JCP, 2023. paper

   Wenhan Gao and Chunmei Wang.

4. An extreme learning machine-based method for computational PDEs in higher dimensions. arXiv, 2023. paper

   Yiran Wang and Suchuan Dong.

5. Multi-resolution active learning of Fourier neural operators. arXiv, 2023. paper

   Shibo Li, Xin Yu, Wei Xing, Mike Kirby, Akil Narayan, and Shandian Zhe.

6. A foundational neural operator that continuously learns without forgetting. arXiv, 2023. paper

   Tapas Tripura and Souvik Chakraborty.

7. Neural Galerkin schemes with active learning for high-dimensional evolution equations. JCP, 2023. paper

   Joan Bruna, Benjamin Peherstorfer, and Eric Vanden-Eijnden.

Applications

Optimization

1. Fast PDE-constrained optimization via self-supervised operator learning. arXiv, 2021. paper

   Sifan Wang, Mohamed Aziz Bhouri, and Paris Perdikaris.

2. An extended physics informed neural network for preliminary analysis of parametric optimal control problems. arXiv, 2021. paper

   Nicola Demo, Maria Strazzullo, and Gianluigi Rozza.

3. Optimal control of PDEs using physics-informed neural networks. JCP, 2023. paper

   Saviz Mowlavi and Saleh Nabi.

4. Solving PDE-constrained control problems using operator learning. AAAI, 2022. paper

   Rakhoon Hwang, Jae Yong Lee, Jin Young Shin, and Hyung Ju Hwang.

5. PDE-based optimal strategy for unconstrained online learning. ICML, 2022. paper

   Zhiyu Zhang, Ashok Cutkosky, and Ioannis Paschalidis.

6. Control of partial differential equations via physics-informed neural networks. Journal of Optimization Theory and Applications, 2022. paper

   Carlos J. García-Cervera, Mathieu Kessler, and Francisco Periago.

7. A machine learning framework for solving high-dimensional mean field game and mean field control problems. PNAS, 2020. paper

   Lars Ruthottoa, Stanley J. Osherc, Wuchen Lic, Levon Nurbekyanc, and Samy Wu Fung.

8. Bi-level physics-informed neural networks for PDE constrained optimization using Broyden's hypergradients. ICLR, 2023. paper

   Zhongkai Hao, Chengyang Ying, Hang Su, Jun Zhu, Jian Song, and Ze Cheng.

9. A combination technique for optimal control problems constrained by random PDEs. arXiv, 2022. paper

   Fabio Nobile and Tommaso Vanzan.

10. A multilevel reinforcement learning framework for PDE-based control. arXiv, 2022. paper

    Atish Dixit and Ahmed H. Elsheikh.

11. Optimal learning of high-dimensional classification problems using deep neural networks. arXiv, 2022. paper

    Philipp Petersen and Felix Voigtlaender.

12. The ADMM-PINNs algorithmic framework for nonsmooth PDE-constrained optimization: A deep learning approach. arXiv, 2023. paper

    Yongcun Song, Xiaoming Yuan, and Hangrui Yue.

13. Learning differentiable solvers for systems with hard constraints. ICLR, 2023. paper

    Geoffrey Négiar, Geoffrey_Négiar, Michael W. Mahoney, and Aditi Krishnapriyan.

14. Volumetric optimal transportation by fast Fourier transform. ICLR, 2023. paper

    Na Lei, DONGSHENG An, Min Zhang, Xiaoyin Xu, and David Gu.

15. PDE-based optimal strategy for unconstrained online learning. ICML, 2022. paper

    Zhiyu Zhang, Ashok Cutkosky, and Ioannis Paschalidis.

16. PDE-constrained models with neural network terms: Optimization and global convergence. JCP, 2023. paper

    Justin Sirignano, Jonathan MacArt, and Konstantinos Spiliopoulos.

17. Topology optimization using neural networks with conditioning field initialization for improved efficiency. arXiv, 2023. paper

    Hongrui Chen, Aditya Joglekar, and Levent Burak Kara.

18. Deep reinforcement learning for optimal well control in subsurface systems with uncertain geology. JCP, 2023. paper

    Yusuf Nasir and Louis J. Durlofsky.

19. Constrained optimization via exact augmented lagrangian and randomized iterative sketching. ICML, 2023. paper

    Ilgee Hong, Sen Na, Michael W. Mahoney, and Mladen Kolar.

20. Efficient PDE-constrained optimization under high-dimensional uncertainty using derivative-informed neural operators. arXiv, 2023. paper

    Dingcheng Luo, Thomas O'Leary-Roseberry, Peng Chen, and Omar Ghattas.

21. Dimension-independent certified neural network watermarks via mollifier smoothing. ICML, 2023. paper

    Jiaxiang Ren, Yang Zhou, Jiayin Jin, Lingjuan Lyu, and Da Yan.

22. Accelerated primal-dual methods with enlarged step sizes and operator learning for nonsmooth optimal control problems. arXiv, 2023. paper

    Yongcun Song, Xiaoming Yuan, and Hangrui Yue.

23. Accelerated primal-dual methods with enlarged step sizes and operator learning for nonsmooth optimal control problems. arXiv, 2023. paper

    Yongcun Song, Xiaoming Yuan, and Hangrui Yue.

24. Operator learning for continuous spatial-temporal model with a hybrid optimization scheme. arXiv, 2023. paper

    Chuanqi Chen and Jinlong Wu.

25. Deep neural operators as accurate surrogates for shape optimization. EAAI, 2023. paper

    Khemraj Shukla, Vivek Oommen, Ahmad Peyvan, Michael Penwarden, Nicholas Plewacki, Luis Bravo, Anindya Ghoshal, Robert M. Kirby, and George Em Karniadakis.

26. Accelerating Bayesian optimal experimental design with derivative-informed neural operators. arXiv, 2023. paper

    Jinwoo Go and Peng Chen.

27. A supervised learning scheme for computing Hamilton-Jacobi equation via density coupling. arXiv, 2024. paper

    Jianbo Cui, Shu Liu, and Haomin Zhou.

Fluid

1. Physics-informed neural networks (PINNs) for fluid mechanics: A review. Acta Mechanica Sinica, 2021. paper

   Shengze Cai, Zhiping Mao, Zhicheng Wang, Minglang Yin, and George Em Karniadakis.

2. Neural operator prediction of linear instability waves in high-speed boundary layers. JCP, 2022. paper

   Patricio Clark Di Leoni, Lu Lu, Charles Meneveau, George Karniadakis, and Tamer A. Zaki.

3. A physics-informed convolutional neural network for the simulation and prediction of two-phase darcy flows in heterogeneous porous media. JCP, 2023. paper

   Zhao Zhang, Xia Yan, Piyang Liu, Kai Zhang, Renmin Han, and Sheng Wang.

4. DiscretizationNet: A machine-learning based solver for Navier–Stokes equations using finite volume discretization. Computer Methods in Applied Mechanics and Engineering, 2021. paper

   Rishikesh Ranade, Chris Hillb, and Jay Pathak.

5. Surrogate modeling for fluid flows based on physics-constrained deep learning without simulation data. Computer Methods in Applied Mechanics and Engineering, 2020. paper

   Luning Sun, Han Gao, Shaowu Pan, and Jianxun Wang.

6. Towards physics-informed deep learning for turbulent flow prediction. KDD, 2020. paper

   Rui Wang, Karthik Kashinath, Mustafa Mustafa, Adrian Albert, and Rose Yu.

7. Learning to estimate and refine fluid motion with physical dynamics. ICML, 2022. paper

   Mingrui Zhang, Jianhong Wang, James Tlhomole, and Matthew D. Piggott.

8. Physics informed neural fields for smoke reconstruction with sparse data. ACM Transactions on Graphics, 2022. paper

   Mengyu Chu, Lingjie Liu, Quan Zheng, Erik Franz, Hans-Peter Seidel, Christian Theobalt, and Rhaleb Zayer.

9. Physics-informed deep learning for traffic state estimation: A hybrid paradigm informed by second-order traffic models. AAAI, 2021. paper

   Rongye Shi, Zhaobin Mo, and Xuan Di.

10. Residual-based adaptivity for two-phase flow simulation in porous media using physics-informed neural networks. Computer Methods in Applied Mechanics and Engineering, 2022. paper

    John M.Hanna, José V.Aguado, Sebastien Comas-Cardona, Ramz Askri, and Domenico Borzacchiello.

11. Learned turbulence modelling with differentiable fluid solvers: Physics-based loss-functions and optimisation horizons. JFM, 2022. paper

    Björn List, Liwei Chen, and Nils Thuerey.

12. Learning hydrodynamic equations for active matter from particle simulations and experiments. PNAS, 2023. paper

    Rohit Supekar, Boya Song, Alasdair Hastewell, Gary P. T. Choi, Alexander Mietke, and Jörn Dunkel.

13. Physics informed neural networks: A case study for gas transport problems. JCP, 2023. paper

    Erik Laurin Strelow, Alf Gerisch, Jens Lang, and Marc E. Pfetsch.

14. Turbulence model augmented physics informed neural networks for mean flow reconstruction. arXiv, 2023. paper

    Yusuf Patel, Vincent Mons, Olivier Marquet, and Georgios Rigas.

15. RANS-PINN based simulation surrogates for predicting turbulent flows. arXiv, 2023. paper

    Shinjan Ghosh, Amit Chakraborty, Georgia Olympia Brikis, and Biswadip Dey.

16. Meta-learning for airflow simulations with graph neural networks. arXiv, 2023. paper

    Wenzhuo Liu, Mouadh Yagoubi, and Marc Schoenauer.

17. Learning operators for identifying weak solutions to the Navier-Stokes equations. arXiv, 2023. paper

    Dixi Wang and Cheng Yu.

18. Physics-informed neural networks modeling for systems with moving immersed boundaries: Application to an unsteady flow past a plunging foil. arXiv, 2023. paper

    Rahul Sundar, Dipanjan Majumdar, Didier Lucor, and Sunetra Sarkar.

19. A machine learning pressure emulator for hydrogen embrittlement. ICML, 2023. paper

    Minh Triet Chau, João Lucas de Sousa Almeida, Elie Alhajjar, and Alberto Costa Nogueira Junior.

20. A probabilistic, data-driven closure model for RANS simulations with aleatoric, model uncertainty. arXiv, 2023. paper

    Atul Agrawal and Phaedon-Stelios Koutsourelakis.

21. Physics-informed machine learning for calibrating macroscopic traffic flow models. arXiv, 2023. paper

    Yu Tang, Li Jin, and Kaan Ozbay.

22. Radial basis function-differential quadrature-based physics-informed neural network for steady incompressible flows. Physics of Fluids, 2023. paper

    Yang Xiao, Liming Yang, Yinjie Du, Yuxin Song, and Chang Shu.

23. Long-term predictions of turbulence by implicit U-Net enhanced Fourier neural operator. Physics of Fluids, 2023. paper

    Zhijie Li, Wenhui Peng, Zelong Yuan, and Jianchun Wang.

24. Physics-informed neural networks for parametric compressible Euler equations. arXiv, 2023. paper

    Simon Wassing, Stefan Langer, and Philipp Bekemeyer.

25. Simulation of rarefied gas flows using physics-informed neural network combined with discrete velocity method. Physics of Fluids, 2023. paper

    Linying Zhang, Wenjun Ma, Qin Lou, and Jun Zhang.

26. Influence of adversarial training on super-resolution turbulence models. arXiv, 2023. paper

    Ludovico Nista, Christoph David Karl Schumann, Mathis Bode, Temistocle Grenga, Jonathan F. MacArt, Antonio Attili, and Heinz Pitsch.

27. Turbulent flow simulation using autoregressive conditional diffusion models. arXiv, 2023. paper

    Georg Kohl, Liwei Chen, and Nils Thuerey.

28. Physics-informed neural networks for studying heat transfer in porous media. International Journal of Heat and Mass Transfer, 2023. paper

    Jiaxuan Xu, Han Wei, and Hua Bao.

29. Solution multiplicity and effects of data and eddy viscosity on Navier-Stokes solutions inferred by physics-informed neural networks. arXiv, 2023. paper

    Zhicheng Wang, Xuhui Meng, Xiaomo Jiang, Hui Xiang, and George Em Karniadakis.

30. Towards real-time training of physics-informed neural networks: Applications in ultrafast ultrasound blood flow imaging. arXiv, 2023. paper

    Haotian Guan, Jinping Dong, and Weining Lee.

31. Multi-physical predictions in electro-osmotic micromixer by auto-encoder physics-informed neural networks. Physics of Fluids, 2023. paper

    Naiwen Chang, Ying Huai, Tingting Liu, Xi Chen, and Yuqi Jin.

32. Studying turbulent flows with physics-informed neural networks and sparse data. International Journal of Heat and Fluid Flow, 2023. paper

    S. Hanrahan, M. Kozul, and R.D. Sandberg.

33. Enhancing physics informed neural networks for solving Navier–Stokes equations. International Journal for Numerical Methods in Fluids, 2023. paper

    Ayoub Farkane, Mounir Ghogho, Mustapha Oudani, and Mohamed Boutayeb.

34. Physics-informed tensor basis neural network for turbulence closure modeling. arXiv, 2023. paper

    Leon Riccius, Atul Agrawal, and Phaedon-Stelios Koutsourelakis.

35. Investigation of low and high-speed fluid dynamics problems using physics-informed neural network. International Journal of Computational Fluid Dynamics, 2023. paper

    Anubhav Joshi,Alexandros Papados, and Rakesh Kumar.

36. Singular layer physics informed neural network method for plane parallel flows. arXiv, 2023. paper

    Tengyuan Chang, Gungmin Gie, Youngjoon Hong, and Chang-Yeol Jung.

37. Data-efficient operator learning for solving high Mach number fluid flow problems. arXiv, 2023. paper

    Noah Ford, Victor J. Leon, Honest Merman, Jeffrey Gilbert, and Alexander New.

38. Three-dimensional laminar flow using physics informed deep neural networks featured. Physics of Fluids, 2023. paper

    Saykat Kumar Biswas and N. K. Anand.

39. Real-time prediction of gas flow dynamics in diesel engines using a deep neural operator framework. Applied Intelligence, 2023. paper

    Varun Kumar, Somdatta Goswami, Daniel Smith, and George Em Karniadakis.

40. Semi-analytic physics informed neural network for convection-dominated boundary layer problems in 2D. arXiv, 2023. paper

    Gungmin Gie, Youngjoon Hong, Chang-Yeol Jung, and Dongseok Lee.

41. The improved backward compatible physics-informed neural networks for reducing error accumulation and applications in data-driven higher-order rogue waves. arXiv, 2023. paper

    Shuning Lin and Yong Chen.

42. Learning characteristic parameters and dynamics of centrifugal pumps under multi-phase flow using physics-informed neural networks. arXiv, 2023. paper

    Felipe de Castro Teixeira Carvalho, Kamaljyoti Nath, Alberto Luiz Serpa, and George Em Karniadakis.

43. On the locality of local neural operator in learning fluid dynamics. arXiv, 2023. paper

    Ximeng Ye, Hongyu Li, Jingjie Huang, and Guoliang Qin.

44. Multi-viscosity physics-informed neural networks for generating ultra high resolution flow field data. International Journal of Computational Fluid Dynamics, 2023. paper

    Sen Zhang, Xiaowei Guo, Chao Li, Ran Zhao, Canqun Yang, Wei Wang, and Yanxu Zhong.

45. Towards high-accuracy deep learning inference of compressible flows over aerofoils. Computers & Fluids, 2023. paper

    Liwei Chen and Nils Thuerey.

46. Data-driven and physics-informed deep learning operators for solution of heat conduction equation with parametric heat source. International Journal of Heat and Mass Transfer, 2023. paper

    Seid Koric and Diab W. Abueidda.

47. Energy-preserving reduced operator inference for efficient design and control. arXiv, 2024. paper

    Tomoki Koike and Elizabeth Qian.

48. Variable linear transformation improved physics-informed neural networks to solve thin-layer flow problems. JCP, 2024. paper

    Jiahao Wu, Yuxin Wu, Guihua Zhang, and Yang Zhang.

49. Riemannonets: Interpretable neural operators for Riemann problems. arXiv, 2024. paper

    Ahmad Peyvan, Vivek Oommen, Ameya D. Jagtap, and George Em Karniadakis.

50. Physics-informed neural networks for incompressible flows with moving boundaries. Physics of Fluids, 2024. paper

    Yongzheng Zhu, Weizhen Kong, Jian Deng, and Xin Bian.

51. Solving the one dimensional vertical suspended sediment mixing equation with arbitrary eddy diffusivity profiles using temporal normalized physics-informed neural networks. Physics of Fluids, 2024. paper

    Shaotong Zhang, Jiaxin Deng, Xi'an Li, Zixi Zhao, Jinran Wu, Weide Li,Yougan Wang, and Dongsheng Jeng.

Cybernetics

1. Machine learning accelerated PDE backstepping observers. arXiv, 2022. paper

   Yuanyuan Shi, Zongyi Li, Huan Yu, Drew Steeves, Anima Anandkumar, and Miroslav Krstic.

2. Neural solvers for fast and accurate numerical optimal control. NIPS, 2021. paper

   Federico Berto, Stefano Massaroli, Michael Poli, and Jinkyoo Park.

3. Bellman neural networks for the class of optimal control problems with integral quadratic cost. TAI, 2022. paper

   Enrico Schiassi, Andrea D'Ambrosio, and Roberto Furfaro.

4. Offline supervised learning vs online direct policy optimization: A comparative study and a unifie training paradigm for neural network-based optimal feedback control. arXiv, 2022. paper

   Yue Zhao and Jiequn Han.

5. Policy evaluation and temporal–difference learning in continuous time and space: A martingale approach. JMLR, 2022. paper

   Yanwei Jia and Xunyu Zhou.

6. Physics-informed kernel embeddings: Integrating prior system knowledge with data-driven control. arXiv, 2023. paper

   Adam J. Thorpe, Cyrus Neary, Franck Djeumou, Meeko M. K. Oishi, and Ufuk Topcu.

7. Distributed control of partial differential equations using convolutional reinforcement learning. arXiv, 2023. paper

   Sebastian Peitz, Jan Stenner, Vikas Chidananda, Oliver Wallscheid, Steven L. Brunton, and Kunihiko Taira.

8. Neural control of parametric solutions for high-dimensional evolution PDEs. arXiv, 2023. paper

   Nathan Gaby, Xiaojing Ye, and Haomin Zhou.

9. Bridging physics-informed neural networks with reinforcement learning: Hamilton-Jacobi-Bellman proximal policy optimization (HJBPPO). arXiv, 2023. paper

   Amartya Mukherjee and Jun Liu.

10. AONN: An adjoint-oriented neural network method for all-at-once solutions of parametric optimal control problems. arXiv, 2023. paper

    Pengfei Yin, Guangqiang Xiao, Kejun Tang, and Chao Yang.

11. Neural operators for bypassing gain and control computations in PDE backstepping. arXiv, 2023. paper

    Luke Bhan, Yuanyuan Shi, and Miroslav Krstic.

12. Neural operators of backstepping controller and observer gain functions for reaction-diffusion PDEs. arXiv, 2023. paper

    Miroslav Krstic, Luke Bhan, and Yuanyuan Shi.

13. Leveraging multi-time Hamilton-Jacobi PDEs for certain scientific machine learning problems. arXiv, 2023. paper

    Paula Chen, Tingwei Meng, Zongren Zou, Jérôme Darbon, and George Em Karniadakis.

14. Learning to control PDEs with differentiable physics. ICLR, 2020. paper

    Philipp Holl, Nils Thuerey, and Vladlen Koltun.

15. A generalizable physics-informed learning framework for risk probability estimation. L4DC, 2020. paper

    Zhuoyuan Wang and Yorie Nakahira.

16. Operator learning for nonlinear adaptive control. L4DC, 2023. paper

    Luke Bhan, Yuanyuan Shi and Miroslav Krstic.

17. Optimal temperature trajectory for tubular reactor using physics informed neural networks. JCP, 2023. paper

    Rahul Patel, Sharad Bhartiya, and Ravindra Gudi.

18. Physics-informed recurrent neural network modeling for predictive control of nonlinear processes. Journal of Process Control, 2023. paper

    Yingzhe Zheng, Cheng Hu, Xiaonan Wang, and Zhe Wu.

19. **Physics-guided neural networks for inversion-based feedforward control applied to hybrid stepper motors.**arXiv, 2023. paper

    Daiwei Fan, Max Bolderman, Sjirk Koekebakker, Hans Butler, and Mircea Lazar.

20. Physics-informed recurrent neural network modeling for predictive control of nonlinear processes. JCP, 2023. paper

    Yingzhe Zheng, Cheng Hu, Xiaonan Wang, and Zhe Wu.

21. Optimal Dirichlet boundary control by Fourier neural operators applied to nonlinear optics. arXiv, 2023. paper

    Nils Margenberg, Franz X. Kärtner, and Markus Bause.

22. Neural operators for delay-compensating control of hyperbolic PIDEs. arXiv, 2023. paper

    Jie Qi, Jing Zhang, and Miroslav Krstic.

23. Physics-informed online learning of gray-box models by moving horizon estimation. European Journal of Control, 2023. paper

    Kristoffer Fink Løwenstein, Daniele Bernardini, Lorenzo Fagiano, and Alberto Bemporad.

24. Online identification and control of PDEs via reinforcement learning methods. arXiv, 2023. paper

    Alessandro Alla, Agnese Pacifico, Michele Palladino, and Andrea Pesare.

25. The hard-constraint PINNs for interface optimal control problems. arXiv, 2023. paper

    Ming-Chih Lai, Yongcun Song, Xiaoming Yuan, Hangrui Yue, and Tianyou Zeng.

26. Deep learning of delay-compensated backstepping for reaction-diffusion PDEs. arXiv, 2023. paper

    Shanshan Wang, Mamadou Diagne, and Miroslav Krstić.

27. Computationally efficient data-driven discovery and linear representation of nonlinear systems for control. arXiv, 2023. paper

    Madhur Tiwari, George Nehma, and Bethany Lusch.

28. Physics-informed state-space neural networks for transport phenomena. arXiv, 2023. paper

    Akshay J Dave and Richard B. Vilim.

29. A comparison of mesh-free differentiable programming and data-driven strategies for optimal control under PDE constraints. arXiv, 2023. paper

    Roussel Desmond Nzoyem, David A.W. Barton, and Tom Deakin.

30. A data-driven tracking control framework using physics-informed neural networks and deep reinforcement learning for dynamical systems. JCP, 2023. paper

    R.R. Faria, B.D.O. Capron, A.R. Secchi, and M.B. De Souza Jr.

31. Leveraging Hamilton-Jacobi PDEs with time-dependent Hamiltonians for continual scientific machine learning. L4DC, 2023. paper

    Paula Chen, Tingwei Meng, Zongren Zou, Jérôme Darbon, and George Em Karniadakis.

32. Physics-informed neural network Lyapunov functions: PDE characterization, learning, and verification. arXiv, 2023. paper

    Jun Liu, Yiming Meng, Maxwell Fitzsimmons, and Ruikun Zhou.

33. Taming waves: A physically-interpretable machine learning framework for realizable control of wave dynamics. arXiv, 2023. paper

    Tristan Shah, Feruza Amirkulova, and Stas Tiomkin.

34. Neural operators for boundary stabilization of stop-and-go traffic. arXiv, 2023. paper

    Yihuai Zhang, Ruiguo Zhong, and Huan Yu.

35. Neural operator approximations of backstepping kernels for 2×2 hyperbolic PDEs. arXiv, 2023. paper

    Shanshan Wang, Mamadou Diagne, and Miroslav Krstić.

36. Lyapunov-based physics-informed long short-term memory (LSTM) neural network-based adaptive control. IEEE Control Systems Letters, 2023. paper

    Rebecca G. Hart, Emily J. Griffis, Omkar Sudhir Patil, and Warren E. Dixon.

37. Gain scheduling with a neural operator for a transport PDE with nonlinear recirculation. arXiv, 2024. paper

    Maxence Lamarque, Luke Bhan, Rafael Vazquez, and Miroslav Krstic.

38. Physics-informed deep learning approach to solve optimal control problem. AIAA, 2024. paper

    Kyung-Mi Na and Chang-Hun Lee.

39. Adaptive neural-operator backstepping control of a benchmark hyperbolic PDE. arXiv, 2024. paper

    Maxence Lamarque, Luke Bhan, Yuanyuan Shi, and Miroslav Krstic.

40. Physical-informed neural network for MPC-based trajectory tracking of vehicles with noise considered. TIV, 2024. paper

    Long Jin, Longqi Liu, Xingxia Wang, Mingsheng Shang, and Feiyue Wang.

Climate

1. FourCastNet: A global data-driven high-resolution weather model using adaptive Fourier neural operators. arXiv, 2022. paper

   Jaideep Pathak, Shashank Subramanian, Peter Harrington, Sanjeev Raja, Ashesh Chattopadhyay, Morteza Mardani, Thorsten Kurth, David Hall, Zongyi Li, Kamyar Azizzadenesheli, Pedram Hassanzadeh, Karthik Kashinath, and Animashree Anandkumar.

2. Fourier neural operators for arbitrary resolution climate data downscaling. JMLR, 2023. paper

   Qidong Yang, Alex Hernandez-Garcia, Paula Harder, Venkatesh Ramesh, Prasanna Sattegeri, Daniela Szwarcman, Campbell D. Watson, and David Rolnick.

3. Modelling atmospheric dynamics with spherical Fourier neural operators. ICLR, 2023. paper

   Boris Bonev, Thorsten Kurth, Christian Hundt, Jaideep Pathak, Maximilian Baust, Karthik Kashinath, and Anima Anandkumar.

4. Spatiotemporal modeling of European paleoclimate using doubly sparse Gaussian processes. NIPS, 2022. paper

   Seth D. Axen, Alexandra Gessner, Christian Sommer, Nils Weitzel, and Álvaro Tejero-Cantero.

5. ClimSim: An open large-scale dataset for training high-resolution physics emulators in hybrid multi-scale climate simulators. arXiv, 2023. paper

   Sungduk Yu, Walter M. Hannah, Liran Peng, Mohamed Aziz Bhouri, Ritwik Gupta, Jerry Lin, Björn Lütjens, Justus C. Will, Tom Beucler, Bryce E. Harrop, Benjamin R. Hillman, Andrea M. Jenney, Savannah L. Ferretti, Nana Liu, Anima Anandkumar, Noah D. Brenowitz, Veronika Eyring, Pierre Gentine, Stephan Mandt, Jaideep Pathak, Carl Vondrick, Rose Yu, Laure Zanna, Ryan P. Abernathey, Fiaz Ahmed, David C. Bader, Pierre Baldi, Elizabeth A. Barnes, Gunnar Behrens, Christopher S. Bretherton, Julius J. M. Busecke, Peter M. Caldwell, Wayne Chuang, Yilun Han, Yu Huang, Fernando Iglesias-Suarez, Sanket Jantre, Karthik Kashinath, Marat Khairoutdinov, Thorsten Kurth, Nicholas J. Lutsko, Po-Lun Ma, Griffin Mooers, J. David Neelin, David A. Randall, Sara Shamekh, Akshay Subramaniam, Mark A. Taylor, Nathan M. Urban, Janni Yuval, Guang J. Zhang, Tian Zheng, and Michael S. Pritchard.

6. Seismic traveltime simulation for variable velocity models using physics-informed Fourier neural operator. arXiv, 2023. paper

   Chao Song, Tianshuo Zhao, Umair bin Waheed, and Cai Liu.

7. DeepPhysiNet: Bridging deep learning and atmospheric physics for accurate and continuous weather modeling. arXiv, 2024. paper

   Wenyuan Li, Zili Liu, Keyan Chen, Hao Chen, Shunlin Liang, Zhengxia Zou, and Zhenwei Shi.

Mechanics

1. Wavelet neural operator for solving parametric partial differential equations in computational mechanics problems. Computer Methods in Applied Mechanics and Engineering, 2023. paper

   Tapas Tripura and Souvik Chakraborty.

2. Graph neural networks for airfoil design. arXiv, 2023. paper

   Florent Bonnet.

3. Exact Dirichlet boundary physics-informed neural network EPINN for solid mechanics. Computer Methods in Applied Mechanics and Engineering, 2023. paper

   Jiaji Wang, Y.L. Mo, Bassam Izzuddin, and Chul-Woo Kim.

4. Solving multi-material problems in solid mechanics using physics-informed neural networks based on domain decomposition technology. Computer Methods in Applied Mechanics and Engineering, 2023. paper

   Yu Diao, Jianchuan Yang, Ying Zhang, Dawei Zhang, and Yiming Du.

5. A novel key performance analysis method for permanent magnet coupler using physics-informed neural networks. Engineering with Computers, 2023. paper

   Huayan Pu, Bo Tan, Jin Yi, Shujin Yuan, Jinglei Zhao, Ruqing Bai, and Jun Luo.

6. Mechanical characterization and inverse design of stochastic architected metamaterials using neural operators. arXiv, 2023. paper

   Hanxun Jin, Enrui Zhang, Boyu Zhang, Sridhar Krishnaswamy, George Em Karniadakis, and Horacio D. Espinosa.

7. A deep learning energy-based method for classical elastoplasticity. International Journal of Plasticity, 2023. paper

   Junyan He, Diab Abueidda, Rashid Abu Al-Ru, Seid Koric, and Iwona Jasiuk.

8. Physics-informed neural networks for magnetostatic problems on axisymmetric transformer geometries. IEEE Journal of Emerging and Selected Topics in Industrial Electronics, 2023. paper

   Philipp Brendel, Vlad Medvedev, and Andreas Rosskopf.

9. Neural born series operator for biomedical ultrasound computed tomography. arXiv, 2023. paper

   Zhijun Zeng, Yihang Zheng, Youjia Zheng, Yubing Li, Zuoqiang Shi, and He Sun.

10. Learning thermoacoustic interactions in combustors using a physics-informed neural network. arXiv, 2024. paper

    Sathesh Mariappan, Kamaljyoti Nath, and George Em Karniadakis.

11. Flight dynamic uncertainty quantification modeling using physics-informed neural networks. AIAA, 2024. paper

    Nathaniel Michek, Piyush Mehta, and Wade Huebsch.

12. Peridynamic neural operators: A data-driven nonlocal constitutive model for complex material responses. arXiv, 2024. paper

    Siavash Jafarzadeh, Stewart Silling, Ning Liu, Zhongqiang Zhang, and Yue Yu.

13. Stochastic dynamics of aircraft ground taxiing via improved physics-informed neural networks. Nonlinear Dynamics, 2024. paper

    Ying Zhang, Zhengrong Jin, Long Wang, Kaixin Zheng, and Wantao Jia.

14. Damage identification for plate structures using physics-informed neural networks. MSSP, 2024. paper

    Wei Zhou and Yongfeng Xu.

Robotics

1. Hybrid learning of time-series inverse dynamics models for locally isotropic robot motion. RAL, 2022. paper

   Tolga-Can Çallar and Sven Böttger.

2. NTFields: Neural time fields for physics-informed robot motion planning. ICLR, 2023. paper

   Ruiqi Ni and Ahmed H Qureshi.

3. Online parameter estimation using physics-informed deep learning for vehicle stability algorithms. arXiv, 2023. paper

   Kemal Koysuren, Ahmet Faruk Keles, and Melih Cakmakci.

4. A locality-based neural solver for optical motion capture. SIGGRAPH, 2023. paper

   Xiaoyu Pan, Bowen Zheng, Xinwei Jiang, Guanglong Xu, Xianli Gu, Jingxiang Li, Qilong Kou, He Wang, Tianjia Shao, Kun Zhou, and Xiaogang Jin.

5. Approximating high-dimensional minimal surfaces with physics-informed neural networks. arXiv, 2023. paper

   Steven Zhou and Xiaojing Ye.

6. A spatial-temporally adaptive PINN framework for 3D bi-ventricular electrophysiological simulations and parameter inference. MICCAI, 2023. paper

   Yubo Ye, Huafeng Liu, Xiajun Jiang, Maryam Toloubidokhti, and Linwei Wang.

7. Using the Transformer model for physical simulation: An application on transient thermal analysis for 3D printing process simulation. NIPS, 2023. paper

   Qian Chen, Luyang Kong, Florian Dugast, and Albert To.

Physics

1. Dynamic weights enabled physics-informed neural network for simulating the mobility of engineered nano-particles in a contaminated aquifer. NIPS, 2022. paper

   Shikhar Nilabh and Fidel Grandia.

2. Learning two-phase microstructure evolution using neural operators and autoencoder architectures. NPJ Computational Materials, 2022. paper

   Vivek Oommen, Khemraj Shukla, Somdatta Goswami, Rémi Dingreville, and George Em Karniadakis.

3. Predicting glass structure by physics-informed machine learning. NPJ Computational Materials, 2022. paper

   Mikkel L. Bødker, Mathieu Bauchy, Tao Du, John C. Mauro, and Morten M. Smedskjaer.

4. Physics-informed deep learning for solving phonon Boltzmann transport equation with large temperature non-equilibrium. NPJ Computational Materials, 2022. paper

   Ruiyang Li, Jianxun Wang, Eungkyu Lee, and Tengfei Luo.

5. Design of Turing systems with physics-informed neural networks. arXiv, 2022. paper

   Jordon Kho, Winston Koh, Jian Cheng Wong, Pao-Hsiung Chiu, and Chin Chun Ooi.

6. Spatio-temporal super-resolution of dynamical systems using physics-informed deep-learning. AAAI, 2023. paper

   Rajat Arora and Ankit Shrivastava.

7. Rapid seismic waveform modeling and inversion with neural operators. TGRS, 2023. paper

   Yan Yang, Angela F. Gao, Kamyar Azizzadenesheli, Robert W. Clayton, and Zachary E. Ross.

8. Accelerating heat exchanger design by combining Physics-Informed deep learning and transfer learning. Chemical Engineering Science, 2023. paper

   Zhiyong Wu, Bingjian Zhang, Haoshui Yu, Jingzheng Ren, Ming Pan, Chang He, and Qinglin Chen.

9. Energy stable neural network for gradient flow equations. arXiv, 2023. paper

   Ganghua Fan, Tianyu Jin, Yuan Lan, Yang Xiang, and Luchan Zhang.

10. Physics-informed neural network with transfer learning (TL-PINN) based on domain similarity measure for prediction of nuclear reactor transients. Scientific Reports, 2023. paper

    Konstantinos Prantikos, Stylianos Chatzidakis, Lefteri H. Tsoukalas, and Alexander Heifetz.

11. Plasma surrogate modelling using Fourier neural operators. arXiv, 2023. paper

    Vignesh Gopakumar, Stanislas Pamela, Lorenzo Zanisi, Zongyi Li, Ander Gray, Daniel Brennand, Nitesh Bhatia, Gregory Stathopoulos, Matt Kusner, Marc Peter Deisenroth, Anima Anandkumar, JOREK Team, and MAST Team.

12. Training a deep operator network as a surrogate solver for two-dimensional parabolic-equation models. Journal of the Acoustical Society of America, 2023. paper

    Liang Xu, Haigang Zhang, and Minghui Zhang.

13. Grad-Shafranov equilibria via data-free physics informed neural networks. arXiv, 2023. paper

    Byoungchan Jang, Alan A. Kaptanoglu, Rahul Gaur, Shaw Pan, Matt Landreman, and William Dorland.

14. Physics-informed deep learning of rate-and-state fault friction. arXiv, 2023. paper

    Cody Rucker and Brittany A. Erickson.

15. Physics-informed neural networks with embedded analytical models: Inverse design of multilayer dielectric-loaded rectangular waveguide devices. IEEE Transactions on Microwave Theory and Techniques, 2023. paper

    Yinqing Pan, Ren Wang, and Bingzhong Wang.

16. En-DeepONet: An enrichment approach for enhancing the expressivity of neural operators with applications to seismology. Computer Methods in Applied Mechanics and Engineering, 2023. paper

    Ehsan Haghighat, Umair bin Waheed, and George Karniadakis.

17. Solving seismic wave equations on variable velocity models with Fourier neural operator. TGRS, 2023. paper

    Bian Li, Hanchen Wang, Shihang Feng, Xiu Yang, and Youzuo Lin.

18. Calculating quasi-normal modes of Schwarzschild black holes with physics informed neural networks. arXiv, 2024. paper

    Nirmal Patel, Aycin Aykutalp, and Pablo Laguna.

19. Deep neural operator-driven real-time inference to enable digital twin solutions for nuclear energy systems. Scientific Reports, 2024. paper

    Kazuma Kobayashi and Syed Bahauddin Alam.

Image

1. Learning to diffuse: A new perspective to design PDEs for visual analysis. TPAMI, 2016. paper

   Risheng Liu, Guangyu Zhong, Junjie Cao, Zhouchen Lin, Shiguang Shan, and Zhongxuan Luo.

2. Reformulating optical flow to solve image-based inverse problems and quantify uncertainty. TPAMI, 2022. paper

   Aleix Boquet-Pujadas and Jean-Christophe Olivo-Marin.

3. WarpPINN: Cine-MR image registration with physics-informed neural networks. arXiv, 2022. paper

   Pablo Arratia Lopez, Hernan Mella, Sergio Uribe, Daniel E. Hurtado, and Francisco Sahli Costabal.

4. NODE-ImgNet: A PDE-informed effective and robust model for image denoising. arXiv, 2023. paper

   Xinheng Xie, Yue Wu, Hao Nib, and Cuiyu He.

5. Microscopy image reconstruction with physics-informed denoising diffusion probabilistic model. arXiv, 2023. paper

   Rui Li, Gabriel della Maggiora, Vardan Andriasyan, Anthony Petkidis, Artsemi Yushkevich, Mikhail Kudryashev, and Artur Yakimovich.

6. TGM-Nets: A deep learning framework for enhanced forecasting of tumor growth by integrating imaging and modeling. EAAI, 2023. paper

   Qijing Chen, Qi Ye, Weiqi Zhang, He Li, and Xiaoning Zheng.

7. The use of physics-informed neural network approach to image restoration via nonlinear PDE tools. Computers & Mathematics with Applications, 2023. paper

   Neda Namaki, M.R. Eslahchi, and Rezvan Salehi.

8. Personalized predictions of glioblastoma infiltration: Mathematical models, physics-informed neural networks and multimodal scans. arXiv, 2023. paper

   Ray Zirui Zhang, Ivan Ezhov, Michal Balcerak, Andy Zhu, Benedikt Wiestler, Bjoern Menze, and John Lowengrub.

9. Real-time FJ/MAC PDE solvers via tensorized, back-propagation-free optical PINN training. arXiv, 2024. paper

   Yequan Zhao, Xian Xian, Xinling Yu, Ziyue Liu, Zhixiong Chen, Geza Kurczveil, Raymond G. Beausoleil, and Zheng Zhang.

Molecules

1. Symmetry-informed geometric representation for molecules, proteins, and crystalline materials. arXiv, 2023. paper

   Shengchao Liu, Weitao Du, Yanjing Li, Zhuoxinran Li, Zhiling Zheng, Chenru Duan, Zhiming Ma, Omar Yaghi, Anima Anandkumar, Christian Borgs, Jennifer Chayes, Hongyu Guo, and Jian Tang.

2. Solving nonconvex energy minimization problems in martensitic phase transitions with a mesh-free deep learning approach. Computer Methods in Applied Mechanics and Engineering, 2023. paper

   Xiaoli Chen, Phoebus Rosakis, Zhizhang Wu, and Zhiwen Zhang.

3. A physics-guided bi-fidelity Fourier-featured operator learning framework for predicting time evolution of drag and lift coefficients. arXiv, 2023. paper

   Amirhossein Mollaali, Izzet Sahin, Iqrar Raza, Christian Moya, Guillermo Paniagua, and Guang Lin.

4. Mixed form based physics-informed neural networks for performance evaluation of two-phase random materials. EAAI, 2023. paper

   Xiaodan Ren and Xianrui Lyu.

5. Deep learning based solution of nonlinear partial differential equations arising in the process of arterial blood flow. Mathematics and Computers in Simulation, 2023. paper

   Bivas Bhaumik, Soumen De, and Satyasaran Changdar.

6. Physics-informed neural networks for solving dynamic two-phase interface problems. SIAM Journal on Scientific Computing, 2023. paper

   Xingwen Zhu, Xiaozhe Hu, and Pengtao Sun.

7. Rethinking materials simulations: Blending direct numerical simulations with neural operators. arXiv, 2023. paper

   Vivek Oommen, Khemraj Shukla, Saaketh Desai, Remi Dingreville, and George Em Karniadakis.

8. A conservative hybrid physics-informed neural network method for Maxwell-Ampère-Nernst-Planck equations. arXiv, 2023. paper

   Cheng Chang, Zhouping Xin, and Tieyong Zeng.

9. A data-driven physics-constrained deep learning computational framework for solving von Mises plasticity. EAAI, 2023. paper

   Arunabha M. Roy and Suman Guha.

10. Learning stiff chemical kinetics using extended deep neural operators. Computer Methods in Applied Mechanics and Engineering, 2024. paper

    Somdatta Goswami, Ameya D. Jagtap, Hessam Babaee, Bryan T. Susi, and George Em Karniadakis.

11. Deep-learning based parameter identification enables rationalization of battery material evolution in complex electrochemical systems. Journal of Computational Science, 2023. paper

    Ivonne Sgura, Luca Mainetti, Francesco Negro, Maria Grazia Quarta, and Benedetto Bozzini.

12. AI-aided geometric design of anti-infection catheters. Science Advances, 2023. paper

    Tingtao Zhou, Xuan Wan, Daniel Zhengyu Huang, Zongyi Li, Zhiwei Peng, Anima Anandkumar, John F. Brady, Paul W. Sternberg, and Chiara Daraio.

13. Physics-informed deep learning to solve three-dimensional Terzaghi’s consolidation equation: Forward and inverse problems. arXiv, 2024. paper

    Biao Yuan, Ana Heitor, He Wang, and Xiaohui Chen.

14. Solving the discretised multiphase flow equations with interface capturing on structured grids using machine learning libraries. arXiv, 2024. paper

    Boyang Chen, Claire E. Heaney, Jefferson L. M. A. Gomes, Omar K. Matar, and Christopher C. Pain.

Reconstruction

1. Occupancy networks: Learning 3D reconstruction in function space. CVPR, 2019. paper

   Lars Mescheder, Michael Oechsle, Michael Niemeyer, Sebastian Nowozin, and Andreas Geiger.

2. Transfer learning for flow reconstruction based on multifidelity data. AIAA Journal, 2022. paper

   Jiaqing Kou, Chenjia Ning, and Weiwei Zhang.

3. Learning-based state reconstruction for a scalar hyperbolic PDE under noisy lagrangian sensing. L4DC, 2022. paper

   Matthieu Barreau, John Liu, and Karl Henrik Johansson.

Quantum

1. Physics-informed neural networks for quantum eigenvalue problems. IJCNN, 2022. paper

   Henry Jin, Marios Mattheakis, and Pavlos Protopapas.

2. Quantum-inspired tensor neural networks for partial differential equations. arXiv, 2022. paper

   Raj Patel, Chia-Wei Hsing, Serkan Sahin, Saeed S. Jahromi, Samuel Palmer, Shivam Sharma, Christophe Michel, Vincent Porte, Mustafa Abid, Stephane Aubert, Pierre Castellani, Chi-Guhn Lee, Samuel Mugel, and Roman Orus.

3. Quantum Fourier networks for solving parametric PDEs. arXiv, 2023. paper

   Nishant Jain, Jonas Landman, Natansh Mathur, and Iordanis Kerenidis.

4. Q-Flow: Generative modeling for differential equations of open quantum dynamics with normalizing flows. ICML, 2023. paper

   Owen M Dugan, Peter Y. Lu, Rumen Dangovski, Di Luo, and Marin Soljacic.

5. Physics-informed quantum machine learning: Solving nonlinear differential equations in latent spaces without costly grid evaluations. arXiv, 2023. paper

   Annie E. Paine, Vincent E. Elfving, and Oleksandr Kyriienko.

6. Physics-informed quantum machine learning for solving partial differential equations. arXiv, 2023. paper

   Abhishek Setty, Rasul Abdusalamov, and Mikhail Itskov.

Game Theory

1. Approximating discontinuous Nash equilibria values of two-player general-sum differential games. arXiv, 2022. paper

   Lei Zhang, Mukesh Ghimire, Wenlong Zhang, Zhe Xu, and Yi Ren.

2. Solving two-player general-sum games between swarms. arXiv, 2023. paper

   Mukesh Ghimire, Lei Zhang, Wenlong Zhang, Yi Ren, and Zhe Xu.

3. Value approximation for two-player general-sum differential games with state constraints. arXiv, 2023. paper

   Lei Zhang, Mukesh Ghimire, Wenlong Zhang, Zhe Xu, and Yi Ren.

4. Pontryagin neural operator for solving parametric general-sum differential games. arXiv, 2024. paper

   Lei Zhang, Mukesh Ghimire, Zhe Xu, Wenlong Zhang, and Yi Ren.

5. Unsupervised solution operator learning for mean-field games via sampling-invariant parametrizations. arXiv, 2024. paper

   Han Huang and Rongjie Lai .

Industry

1. Physics-aware machine learning surrogates for real-time manufacturing digital twin. Manufacturing Letters, 2022. paper

   Aditya Balu, Soumik Sarkar, Baskar Ganapathysubramanian, and Adarsh Krishnamurthy.

2. Multi-scale digital twin: Developing a fast and physics-informed surrogate model for groundwater contamination with uncertain climate models. arXiv, 2022. paper

   Lijing Wang, Takuya Kurihana, Aurelien Meray, Ilijana Mastilovic, Satyarth Praveen, Zexuan Xu, Milad Memarzadeh, Alexander Lavin, and Haruko Wainwright.

3. SciAI4Industry--Solving PDEs for industry-scale problems with deep learning. arXiv, 2022. paper

   Philipp A. Witte, Russell J. Hewett, Kumar Saurabh, AmirHossein Sojoodi, and Ranveer Chandra.

4. Operator learning framework for digital twin and complex engineering systems. arXiv, 2023. paper

   Kazuma Kobayashi, James Daniell, and Syed B. Alam.

5. Towards solving industry-grade surrogate modeling problems using physics informed machine learning. arXiv, 2023. paper

   Saakaar Bhatnagar, Andrew Comerford, and Araz Banaeizadeh.

6. Data-driven physics-informed neural networks: A digital twin perspective. arXiv, 2024. paper

   Sunwoong Yang, Hojin Kim, Yoonpyo Hong, Kwanjung Yee, Romit Maulik, and Namwoo Kang.

7. Physically informed synchronic-adaptive learning for industrial systems modeling in heterogeneous media with unavailable time-varying interface. arXiv, 2024. paper

   Aina Wang, Pan Qin, and Ximing Sun.

8. Improved generalization with deep neural operators for engineering systems: Path towards digital twin. EAAI, 2024. paper

   Kazuma Kobayashi, James Daniell, and Syed Bahauddin Alam.

Economics

1. A deep learning based numerical PDE method for option pricing. Computational Economics, 2023. paper

   Xiang Wang, Jessica Li, and Jichun Li.