Thermodynamics-informed graph neural networks

Abstract

In this paper we present a deep learning method to predict the temporal evolution of dissipative dynamic systems. We propose using both geometric and thermodynamic inductive biases to improve accuracy and generalization of the resulting integration scheme. The first is achieved with Graph Neural Networks, which induces a non-Euclidean geometrical prior with permutation invariant node and edge update functions. The second bias is forced by learning the GENERIC structure of the problem, an extension of the Hamiltonian formalism, to model more general non-conservative dynamics. Several examples are provided in both Eulerian and Lagrangian description in the context of fluid and solid mechanics respectively, achieving relative mean errors of less than 3% in all the tested examples. Two ablation studies are provided based on recent works in both physics-informed and geometric deep learning.

For more information, please refer to the following:

• Hernández, Quercus and Badías, Alberto and Chinesta, Francisco and Cueto, Elías. "Thermodynamics-informed graph neural networks." IEEE Transactions on Artificial Intelligence (2022).

Setting it up

First, clone the project.

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

Then, install the needed dependencies. The code is implemented in Pytorch. Note that this has been tested using Python 3.9.

# install dependencies
pip install numpy scipy matplotlib torch torch-geometric torch-scatter

How to run the code

Test pretrained nets

The results of the paper (Couette flow, bending beam and flow past a cylinder) can be reproduced with the following scripts, found in the executables/ folder.

python main.py --sys_name couette --train False --n_hidden 2 --dim_hidden 10 --passes 10
python main.py --sys_name beam --train False --n_hidden 2 --dim_hidden 50 --passes 10
python main.py --sys_name cylinder --train False --n_hidden 2 --dim_hidden 128 --passes 10

The data/ folder includes the database and the pretrained parameters of the networks. The resulting time evolution of the state variables is plotted and saved in .gif format in a generated outputs/ folder.

Couette Flow	Bending Beam	Flow past a Cylinder

Train a custom net

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

e.g.
python main.py --sys_name beam --train True --lr 1e-3 ...

General Arguments:

Argument	Description	Options
--sys_name	Study case	couette, beam, cylinder
--train	Train mode	True, False
--gpu	Enable GPU acceleration	True, False
--output_dir	Output data directory	Default: output
--plot_sim	Plot a test simulation	True, False

Training Arguments:

Argument	Description	Options
--n_hidden	Number of MLP hidden layers	Default: 2
--dim_hidden	Dimension of hidden layers	Default: 10
--passes	Number of message passing blocks	Default: 10
--lr	Learning rate	Default: 1e-3
--lambda_d	Data loss weight	Default: 1e2
--noise_var	Variance of the training noise	Default: 1e-2
--batch_size	Training batch size	Default: 2
--max_epoch	Maximum number of training epochs	Default: 6000
--miles	Learning rate scheduler milestones	Default: 2000 4000
--gamma	Learning rate scheduler decay	Default: 1e-1

Citation

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

@article{hernandez2022thermodynamics,
  title={Thermodynamics-informed Graph Neural Networks},
  author={Hernandez, Quercus and Badias, Alberto and Chinesta, Francisco and Cueto, Elias},
  journal={IEEE Transactions on Artificial Intelligence},
  year={2022},
  publisher={IEEE}
}