Deep learning of thermodynamics-aware reduced-order models from data

Abstract

We present an algorithm to learn the relevant latent variables of a large-scale discretized physical system and predict its time evolution using thermodynamically-consistent deep neural networks. Our method relies on sparse autoencoders, which reduce the dimensionality of the full order model to a set of sparse latent variables with no prior knowledge of the coded space dimensionality. Then, a second neural network is trained to learn the metriplectic structure of those reduced physical variables and predict its time evolution with a so-called structure-preserving neural network. This data-based integrator is guaranteed to conserve the total energy of the system and the entropy inequality, and can be applied to both conservative and dissipative systems. The integrated paths can then be decoded to the original full-dimensional manifold and be compared to the ground truth solution. This method is tested with two examples applied to fluid and solid mechanics.

For more information, please refer to the following:

• Hernández, Quercus and Badías, Alberto and González, David and Chinesta, Francisco and Cueto, Elías. "Deep learning of thermodynamics-aware reduced-order models from data." Computer Methods in Applied Mechanics and Engineering (2021).

Setting it up

First, clone the project.

# clone project
git clone https://github.com/quercushernandez/DeepLearningMOR.git
cd DeepLearningMOR

Then, install the needed dependencies. The code is implemented in Pytorch. Note that this has been tested using Python 3.7.

# install dependencies
pip install scipy numpy matplotlib torch

How to run the code

Test pretrained nets

The results of the paper (Viscoelastic Fluid and Hyperelastic Rolling Tire) can be reproduced with the following executables, found in the executables/ folder. The data/ folder includes the database and the pretrained parameters of the networks.

sh executables/run_viscoelastic_test.sh
sh executables/run_rolling_tire_test.sh

The resulting time evolution of the latent variables (reduced with the SAE) and the thermodynamics-aware integrator (SPNN) are plotted and saved in .png format in a generated outputs/ folder.

Viscoelastic Fluid	Rolling Tire

Train a custom net

You can also run your own experiments for the implemented datasets by setting parameters manually. Several training examples can be found in the executables/ folder. Both the SAE and the SPNN can be trained individually, and the trained parameters and output plots are saved in the outputs/ folder.

e.g.
python main.py --sys_name viscoelastic --train_SAE True --lr_SAE 1e-5 --lambda_r 1e-3 --train_SPNN True --lr_SPNN 1e-5 ...

General Arguments:

Argument	Description	Options
--sys_name	Study case	viscoelastic, rolling_tire
--train_SAE	Train or test mode for SAE	True, False
--train_SPNN	Train or test mode for SPNN	True, False
--dset_dir	Dataset and pretrained nets directory	Default: data
--train_percent	Train porcentage of the full database	Default: 0.8
--output_dir	Output data directory	Default: output
--save_plots	Save plots of latent and state variables	True, False

Sparse Autoencoder (SAE) Arguments:

Argument	Description	Options
--layer_vec_SAE	Layer vector (viscoelastic)	Default: 400, 160, 160, 10
--layer_vec_q_SAE	Layer vector (rolling_tire, position)	Default: 12420, 40, 40, 10
--layer_vec_v_SAE	Layer vector (rolling_tire, velocity)	Default: 12420, 40, 40, 10
--layer_vec_sigma_SAE	Layer vector (rolling_tire, stress tensor)	Default: 24840, 80, 80, 20
--activation_SAE	Activation functions of the hidden layers	Default: 2e3
--max_epoch_SAE	Maximum number of training epochs	Default: 10e3
--lr_SAE	Learning rate	Default: 1e-4
--lambda_r_SAE	Sparse regularizer	Default: 1e-3

Structure-Preserving Neural Network (SPNN) Arguments:

Argument	Description	Options
--hidden_vec_SPNN	Hidden layers vector	Default: 24, 24, 24, 24
--activation_SPNN	Activation functions of the hidden layers	linear, sigmoid, relu, tanh
--init_SPNN	Net initialization method	kaiming_normal, xavier_normal
--max_epoch_SPNN	Maximum number of training epochs	Default: 1e4
--lr_SPNN	Learning rate	Default: 1e-3
--lambda_d	Data weight	Default: 5e2
--miles	Learning rate scheduler milestones	Default: 9e4
--gamma	Learning rate scheduler decay	Default: 1e-1

Citation

If you found this code useful please cite our work as:

@article{hernandez2020deep,
  title={Deep learning of thermodynamics-aware reduced-order models from data},
  author={Hern{\'a}ndez, Quercus and Bad{\'\i}as, Alberto and Gonz{\'a}lez, David and Chinesta, Francisco and Cueto, El{\'\i}as},
  journal={arXiv preprint arXiv:2007.03758},
  year={2020}
}