Physics-informed neural networks for solving differential equations
Differential equations can describe various natural and social phenomena. The physics-informed neural networks (PINNs), as a deep learning framework, is a powerful and effective way in solving forward and inverse problems involving differential equations. The course will review the publications of the recent years on PINNs, including explanation of the principle, numerical examples and codes of vanilla PINN and the various improved methods. At the same time, audiences studying PINNs are encouraged to share their research experience and achievements.
Lecturer
Date
7th March ~ 27th June, 2023
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Tuesday | 13:30 - 16:05 | A3-3-201 | ZOOM 03 | 242 742 6089 | BIMSA |
Prerequisite
Knowledges on mathematical physics equations, deep neural networks and the Python language.
Syllabus
1. Overview, the original reference of PINN, the code of PINN using Tensorflow
2. Domain decomposition: cPINN/XPINN, parallel PINN, adaptive activation function
3. Configuration point sampling method: adaptive refinement, gPINN, systematic study
4. To improve the training process of PINN: bcPINN, Seq2seq/Curriculum learning, Causality
5. The design of weight: adaptive weights, point-weighted, gwPINN
6. Variational framework: basic knowledge review, DGM, Deep Ritz, hp-VPINNs
7. Combination of PINN and discrete numerical scheme
8. Analysis and improvement of the failure of PINN(a): the landscape of the loss function, code, introduction to frequency principle
9. Analysis and improvement of the failure of PINN(b): neural tangent kernel(NTK) theory, Fourier feature embeddings, Multi-scale DNN
10. Solving hyperbolic conservation laws with PINN(a): cvPINN, discrete divergence operator
11. Solving hyperbolic conservation laws with PINN(b): Literature review of solving Euler equations by PINN, application on inverse problem
12. Operator learning(a): Review of basic knowledge of functional analysis, the reference of Chen & Chen (1995), DeepONet
13. Operator learning(b): physics-informed DeepONet, V-DeepONet
14. Operator learning(c): MIONet, DeepM&Mnet, Multifidelity DeepONet
15. Operator learning(d): Fourier neural operator (FNO), comparison of DeepONet and FNO
16. Course review and interactive communication
2. Domain decomposition: cPINN/XPINN, parallel PINN, adaptive activation function
3. Configuration point sampling method: adaptive refinement, gPINN, systematic study
4. To improve the training process of PINN: bcPINN, Seq2seq/Curriculum learning, Causality
5. The design of weight: adaptive weights, point-weighted, gwPINN
6. Variational framework: basic knowledge review, DGM, Deep Ritz, hp-VPINNs
7. Combination of PINN and discrete numerical scheme
8. Analysis and improvement of the failure of PINN(a): the landscape of the loss function, code, introduction to frequency principle
9. Analysis and improvement of the failure of PINN(b): neural tangent kernel(NTK) theory, Fourier feature embeddings, Multi-scale DNN
10. Solving hyperbolic conservation laws with PINN(a): cvPINN, discrete divergence operator
11. Solving hyperbolic conservation laws with PINN(b): Literature review of solving Euler equations by PINN, application on inverse problem
12. Operator learning(a): Review of basic knowledge of functional analysis, the reference of Chen & Chen (1995), DeepONet
13. Operator learning(b): physics-informed DeepONet, V-DeepONet
14. Operator learning(c): MIONet, DeepM&Mnet, Multifidelity DeepONet
15. Operator learning(d): Fourier neural operator (FNO), comparison of DeepONet and FNO
16. Course review and interactive communication
Reference
Published literatures related to PINNs since 2019, which will be informed before each class.
Audience
Undergraduate
, Graduate
Video Public
Yes
Notes Public
Yes
Language
Chinese
Lecturer Intro
Fansheng Xiong (熊繁升) is currently an Assistant Researcher Fellow of BIMSA. Before that, he got a bachelor's degree from China University of Geosciences (Beijing), and a doctoral degree from Tsinghua University. He was a visiting student at Yale University for one year. His research interest mainly focuses on solving PDE-related forward/inverse problems based on machine learning algorithms (DNN, PINN, DeepONet, etc.), and their applications in geophysical wave propagation problems and turbulence modeling of fluid mechanics. The relevant efforts have been published in journals such as JGR Solid Earth, GJI, Geophysics, etc.