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Combining physics-informed graph neural network and finite difference for solving forward and inverse spatiotemporal PDEs

Hao Zhang, Longxiang Jiang, Xinkun Chu, Yong Wen, Luxiong Li, Yonghao Xiao, Liyuan Wang

TL;DR

This work proposes a novel discrete approach termed Physics-Informed Graph Neural Network (PIGNN) to solve forward and inverse nonlinear PDEs, which seamlessly integrates the strength of graph neural networks (GNN), physical equations and finite difference to approximate solutions of physical systems.

Abstract

The great success of Physics-Informed Neural Networks (PINN) in solving partial differential equations (PDEs) has significantly advanced our simulation and understanding of complex physical systems in science and engineering. However, many PINN-like methods are poorly scalable and are limited to in-sample scenarios. To address these challenges, this work proposes a novel discrete approach termed Physics-Informed Graph Neural Network (PIGNN) to solve forward and inverse nonlinear PDEs. In particular, our approach seamlessly integrates the strength of graph neural networks (GNN), physical equations and finite difference to approximate solutions of physical systems. Our approach is compared with the PINN baseline on three well-known nonlinear PDEs (heat, Burgers and FitzHugh-Nagumo). We demonstrate the excellent performance of the proposed method to work with irregular meshes, longer time steps, arbitrary spatial resolutions, varying initial conditions (ICs) and boundary conditions (BCs) by conducting extensive numerical experiments. Numerical results also illustrate the superiority of our approach in terms of accuracy, time extrapolability, generalizability and scalability. The main advantage of our approach is that models trained in small domains with simple settings have excellent fitting capabilities and can be directly applied to more complex situations in large domains.

Combining physics-informed graph neural network and finite difference for solving forward and inverse spatiotemporal PDEs

TL;DR

This work proposes a novel discrete approach termed Physics-Informed Graph Neural Network (PIGNN) to solve forward and inverse nonlinear PDEs, which seamlessly integrates the strength of graph neural networks (GNN), physical equations and finite difference to approximate solutions of physical systems.

Abstract

The great success of Physics-Informed Neural Networks (PINN) in solving partial differential equations (PDEs) has significantly advanced our simulation and understanding of complex physical systems in science and engineering. However, many PINN-like methods are poorly scalable and are limited to in-sample scenarios. To address these challenges, this work proposes a novel discrete approach termed Physics-Informed Graph Neural Network (PIGNN) to solve forward and inverse nonlinear PDEs. In particular, our approach seamlessly integrates the strength of graph neural networks (GNN), physical equations and finite difference to approximate solutions of physical systems. Our approach is compared with the PINN baseline on three well-known nonlinear PDEs (heat, Burgers and FitzHugh-Nagumo). We demonstrate the excellent performance of the proposed method to work with irregular meshes, longer time steps, arbitrary spatial resolutions, varying initial conditions (ICs) and boundary conditions (BCs) by conducting extensive numerical experiments. Numerical results also illustrate the superiority of our approach in terms of accuracy, time extrapolability, generalizability and scalability. The main advantage of our approach is that models trained in small domains with simple settings have excellent fitting capabilities and can be directly applied to more complex situations in large domains.
Paper Structure (20 sections, 1 theorem, 48 equations, 22 figures, 1 table)

This paper contains 20 sections, 1 theorem, 48 equations, 22 figures, 1 table.

Table of Contents

  1. PIGNN Framework
  2. Problem setup
  3. Model architecture
  4. Physics-informed loss function
  5. Rules of selecting features
  6. Some computational issues on the graph
  7. Differential operators on the graph
  8. Time derivative
  9. Gradient operator
  10. Laplace operator
  11. Interpolation method for inverse problem
  12. Experiments
  13. Evaluation metrics
  14. Results
  15. Extrapolability of the forward problem
  16. ...and 5 more sections

Key Result

Lemma 2.2

Let $\,\bm{{\rm x}}_0=\bm{0}$, $\bm{{\rm x}}_1,\cdots,\bm{{\rm x}}_p\in\mathcal{N}(\bm{0}), \,d=2,\, \Psi=\Psi_2=\{x, y, x^2, xy,y^2,\cdots,\\ x^m,\cdots,y^m\}$, $\varphi_1=x,\, \varphi_2=y,\cdots$, $|\Psi|=n$, $\bm{\omega}_r$ be the least square solution of the following linear system where Let $\bm{\eta}_r=r^2\bm{\omega}_r$, then 1) $\bm{\eta}_r$ is bounded. 2) Let $\{\bm{\eta}_n\}_{n=1}^{\inf

Figures (22)

  • Figure 1: The schematic diagram of PIGNN for solving forward and inverse problems. Unstructured meshes are generated using numerical methods. The Encoder component of the network constructs the input graph from mesh data, Processor updates and transmits the information, and Decoder outputs PDE solutions. By applying the finite-difference-based method, we compute all differential terms and incorporate prior knowledge of PDEs into the network via the specific PDE loss.
  • Figure 2: The network architecture of PIGNN.
  • Figure 3: Comparison of time extrapolability between PIGNN and PINN for heat equation. Computational domain: $[0,1]\times[0,1]$, mesh density=100.
  • Figure 4: Comparison of time extrapolability between PIGNN and PINN for Burgers equation. Computational domain: $[0,1]\times[0,1]$, mesh density=100.
  • Figure 5: Time extrapolability of PIGNN compared with numerical results for FN equation. Computational domain: $[0,1]\times[0,1]$, mesh density=100.
  • ...and 17 more figures

Theorems & Definitions (4)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • proof