Autoencoders in a physical-driven problem

Training data

Each dataset is defined by its directory name and is located in the datasets folder.

Each dataset must contain all images for training, validation and testing.

Implemented input formats are:

• PNG images

• TXT files with the following information (in the same directory):

  1. Th.msh file containing the mesh information

     1st column	2n column	3rd column
     nNodes	nElems	nBoundEdges
     x_1	y_1	edgeID_1
     :	:	:
     x_nNodes	y_nNodes	edgeID_nNodes
     n1_1	n2_1	n3_1
     :	:	:
     n1_nElems	n2_nElems	n3_nElems
     e_n1_1	e_n2_1	e_boundID_1
     :	:	:
     e_n1_nBoundEdges	e_n2_nBoundEdges	e_boundID_nBoundEdges

     First, a block of length nNodes with nodal positions x and y for each node in the mesh (x, y, edgeID).

     Second a block of length nElems with connectivity matrix, nodes in each element (node1, node2, node3).

     Thrid a block of length nBoundEdges with information of each boundary edge; (node1, node2, boundID).

  2. fileName.txt files with element values (solution with P0, constant values in each element). This files must be all related to the same mesh in Th.msh

     1st column	2n column	3rd column	...	nth column
     nElems
     u_1	u_2	u_3	u_4	u_5
     :	:	:	:	:
     ...	...	...	...	u_nElems

FCNN model

FCNN architecture for the beam_homog_x4 dataset

• Input layer - 80x160 = 12800 neurons - 0 train params
• Hidden layer - 200 neurons - 200*12800 + 200 = 2560200 train params
• Hidden layer - 200 neurons - 200*200 + 200 = 40200 train params
• Hidden layer - 200 neurons - 200*200 + 200 = 40200 train params
• Hidden layer - 200 neurons - 200*200 + 200 = 40200 train params
• Latent space (code) - max size 25 - 25*200 + 25 = 5025 train params
• Hidden layer - 200 neurons - 200*25 + 200 = 5200 train params
• Hidden layer - 200 neurons - 200*200 + 200 = 40200 train params
• Hidden layer - 200 neurons - 200*200 + 200 = 40200 train params
• Hidden layer - 200 neurons - 200*200 + 200 = 40200 train params
• Output layer - 80x160 = 12800 neurons - 12800*200 + 12800 = 2572800 train params

CNN model

CNN architecture for the beam_homog_x4 dataset

• Input layer - 80x160x1 = 12800 neurons - 0 train params
• Conv layer with 20 filters of kernel 3 - 80x160x20 (each of the 20 filters is 3x3) - 20x9 + 20 = 200 train params
• MaxPol layer with stride 2 - 40x80x20 - 0 train params
• Conv layer with 20 filters of kernel 3 - 40x80x20 (each of the 20 filters is 3x3) - 180x20 + 20 = 3620 train params
• MaxPol layer with stride 2 - 20x40x20 - 0 train params
• Hidden layer - 20x40x20 = 16000 neurons - 0 train params
• Hidden layer - 400 neurons - 400*16000 + 400 = 6400400 train params
• Hidden layer - 400 neurons - 400*400 + 400 = 160400 train params
• Latent space (code) - max size 25 - 25*400 + 25 = 10025 train params
• Hidden layer - 400 neurons = 400*25 + 400 = 10400 train params
• Hidden layer - 400 neurons = 400*400 + 400 = 160400 train params
• Hidden layer - 20x40x20 = 16000 neurons - 16000*400 + 16000 = 6416000 train params
• Transposed conv layer with 20 filters, kernel 3 and stride 2 - 40x80x20 - 20x9x20 + 20 = 3620 train params
• Transposed conv layer with 20 filters, kernel 3 and stride 2 - 80x160x20 - 20x9x20 + 20 = 3620 train params
• Output layer - 80x160x1 = 12800 neurons - 20x9 + 1 = 181 train params

Results

1. "beam_homog_x4" dataset using a FCNN

Training, validation and testing datasets

Building and compiling model

Autoencoder training

Training parameters and loss functions obtained durig training.

Prediction results

Results obtained using the autoencoder.

First row corresponds to the original image and last row are the recevered image after being passed through the autoencoder. The middle row corresponds to the latent space (code) representaion, where the information is compressed preserving the accuracy.

Layer	Resolution	Size	Compression rate
Original image	80x160x1	12800	1
Latent space	5x5	25 (only 7 non-zero)	1800
Reconstructed image	80x160x1	12800	1

2. "beam_simp_x4" dataset using a CNN

Training, validation and testing datasets

Building and compiling model

Autoencoder training

Training parameters and loss functions obtained durig training.

Prediction results

Results obtained using the autoencoder.

First row corresponds to the original image and last row are the recevered image after being passed through the autoencoder. The middle row corresponds to the latent space (code) representaion, where the information is compressed preserving the accuracy.

Layer	Resolution	Size	Compression rate
Original image	80x160x1	12800	1
Latent space	5x5	25 (only 17 non-zero)	750
Reconstructed image	80x160x1	12800	1