Tutorials for Physics-Informed Neural Networks (PINNs)

This repository provides step-by-step guides to Physics-informed neural networks (PINNs).

Part 1: Data-driven machine learning methods: strengths and limits

In this section, we only focus on data-driven machine learning methods. The tutorial shows how these methods approximate the solution of a parial diffrential equation (PDE).

To run online the tutorial:

(recommended, a google account is required)

Part 2: PINNs and their scope of use

In this section, we focus on PINNs. The tutorial shows how to use PINNs to solve different types of problems involving partial differential equation (PDE) or system of PDEs.

To run online the tutorial:

(recommended, a google account is required)

Part 3: Strategies to improve PINNs performance (available soon)

Part 4: Integrate the geometries into PINNs (available soon)

Acknowledgement

This work is funded by Michelin and CEA through the Industrial Data Analytics and Machine Learning chair of Borelli Center, ENS Paris-Saclay.

References

Below is the list of references used in this repository.

[1] Raissi, M., Perdikaris, P. and Karniadakis, G.E., 2019. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics, 378, pp.686-707. link to paper

[2] Lu, L., Meng, X., Mao, Z. and Karniadakis, G.E., 2021. DeepXDE: A deep learning library for solving differential equations. SIAM review, 63(1), pp.208-228. link to paper

[3] McClenny, L.D. and Braga-Neto, U.M., 2023. Self-adaptive physics-informed neural networks. Journal of Computational Physics, 474, p.111722. link o paper

[4] Wang, S., Yu, X. and Perdikaris, P., 2022. When and why PINNs fail to train: A neural tangent kernel perspective. Journal of Computational Physics, 449, p.110768. link to paper

[5] Wang, S., Teng, Y. and Perdikaris, P., 2021. Understanding and mitigating gradient flow pathologies in physics-informed neural networks. SIAM Journal on Scientific Computing, 43(5), pp.A3055-A3081. link to paper

[6] Jagtap, A.D., Kawaguchi, K. and Em Karniadakis, G., 2020. Locally adaptive activation functions with slope recovery for deep and physics-informed neural networks. Proceedings of the Royal Society A, 476(2239), p.20200334. link to paper

[7] Jagtap, A.D., Shin, Y., Kawaguchi, K. and Karniadakis, G.E., 2022. Deep Kronecker neural networks: A general framework for neural networks with adaptive activation functions. Neurocomputing, 468, pp.165-180. link to paper

[8] Nguyen, T.N.K., Dairay, T., Meunier, R., Millet, C. and Mougeot, M., 2023, June. Fixed-Budget Online Adaptive Learning for Physics-Informed Neural Networks. Towards Parameterized Problem Inference. In International Conference on Computational Science (pp. 453-468). Cham: Springer Nature Switzerland. link to paper

[9] Wu, C., Zhu, M., Tan, Q., Kartha, Y. and Lu, L., 2023. A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks. Computer Methods in Applied Mechanics and Engineering, 403, p.115671. link to paper

[10] Daw, A., Bu, J., Wang, S., Perdikaris, P. and Karpatne, A., 2022. Rethinking the importance of sampling in physics-informed neural networks. arXiv preprint arXiv:2207.02338. link to paper

[11] Wang, S., Sankaran, S. and Perdikaris, P., 2022. Respecting causality is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404. link to paper

[12] Nguyen, T.N.K., Dairay, T., Meunier, R. and Mougeot, M., 2022. Physics-informed neural networks for non-Newtonian fluid thermo-mechanical problems: An application to rubber calendering process. Engineering Applications of Artificial Intelligence, 114, p.105176. link to paper

drpengyuhuang/pinns_tutorial