This project is a replication of ''Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations'' by M. Raissi, P. Perdikaris, and G.E. Karniadakis from 2019.

Full reference to the original article :

Raissi, M., P. Perdikaris, and G. E. Karniadakis. “Physics-Informed Neural Networks: A Deep Learning Framework for Solving Forward and Inverse Problems Involving Nonlinear Partial Differential Equations.” Journal of Computational Physics 378 (February 1, 2019): 686–707. https://doi.org/10.1016/j.jcp.2018.10.045.

GitHub Repository of original work:

https://github.com/maziarraissi/PINNs

The aim of this repository was to:

• Reproduce the figures from the main manuscript of Raissi et al. (2019), originally obtained with Tensorflow 1x, using the Python library PyTorch.
• Save the models obtained from the training.
• Record the training information such as computing times and the accuracies achieved.

main/:

• Data/: Contains .mat files with the required inputs for the models.

• continuous_time_inference (Schrodinger)/: Results in Figure 1, corresponding to the the 3.1.1. Example (Schrodinger equation).

• discrete_time_inference (AC)/: Results in Figure 2, corresponding the the 3.2.1. Example (Allen–Cahn equation).

• continuous_time_identification (Navier-Stokes)/: Results in Figure 4, corresponding the the 4.1.1. Example (Navier–Stokes equation).

• discrete_time_identification (KdV)/: Results in Figure 5, corresponding the the 4.2.1. Example (Korteweg–de Vries equation).

appendix/:

• Data/: Contains .mat files with the required inputs for the models.

• continuous_time_inference (Burgers)/: Results in Figure A.6, corresponding to the the A.1. Continuous time models.

• discrete_time_inference (Burgers)/: Results in Figure A.7, corresponding to the the A.7. Discrete time models.

• continuous_time_identification (Burgers)/: Results in Figure B.8, corresponding to the the B.2. Discrete time models.

• discrete_time_identification (Burgers)/: Results in Figure B.9, corresponding to the the B.3. Discrete time models.

Each example contains the main and plotting codes, figures (figures/), model (.pt) and summary information about the training process (training/).

We recommend setting up a new Python environment with conda. You can do this by running the following commands:

conda env create -f environment.yml
conda activate ReScience-PINNs-env

To verify the packages installed in your ReScience-PINNs-env conda environment, you can use the following command:

conda list -n ReScience-PINNs-env

Run the scripts individually:

Schrodinger Equation - Continuous time inference

make run_Schrodinger_main

make run_Schrodinger_plots

AC Equation - Discrete time inference

make run_AC_main

make run_AC_plots

NS equation - clean and noisy data - Continuous time identification

make run_NS_clean_main

make run_NS_noisy_main

make run_NS_plots

kdV equation - clean and noisy data - Discrete time identification

make run_kdV_clean_main

make run_kdV_noisy_main

make run_kdV_plots

Burgers equation - Continuous time inference

make run_Burgers_ctin_main

make run_Burgers_ctin_plots

make run_Burgers_ctin_main_systematic

Burgers equation - Discrete time inference

make run_Burgers_dtin_main

make run_Burgers_dtin_plots

make run_Burgers_dtin_main_systematic

Burgers equation - Continuous time identification

make run_Burgers_ctid_main

make run_Burgers_ctid_plots

make run_Burgers_ctid_main_systematic

Burgers equation - Discrete time identification

make run_Burgers_dtid_main

make run_Burgers_dtid_plots

make run_Burgers_dtid_main_systematic

Run all scripts in sequence:

make all

The models were trained with a NVIDIA GeForce RTX A2000 GPU card. The summary of the training information such as the computing times is included in the the folder of each simulation.

• Schrodinger_main.py
• Schrodinger_plots.py
• AC_main.py
• AC_plots.py
• NS_clean_main.py
• NS_noisy_main.py
• NS_plots.py
• kdV_clean_main.py
• kdV_noisy_main.py
• kdV_plots.py
• Burgers_ctin_main.py
• Burgers_ctin_plots.py
• Burgers_ctin_main_systematic.py
• Burgers_dtin_main.py
• Burgers_dtin_plots.py
• Burgers_dtin_main_systematic.py
• Burgers_ctid_clean_main.py
• Burgers_ctid_noisy_main.py
• Burgers_ctid_plots.py
• Burgers_ctid_main_systematic.py
• Burgers_dtid_clean_main.py
• Burgers_dtid_noisy_main.py
• Burgers_dtid_plots.py
• Burgers_dtid_main_systematic.py