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

TL;DR

This paper introduces a physics-informed graph neural network method that effectively solves forward and inverse nonlinear PDEs, demonstrating superior accuracy, scalability, and adaptability over traditional PINNs, especially on irregular meshes and complex conditions.

Contribution

The paper presents a novel discrete GNN-based approach integrating finite difference methods for solving nonlinear PDEs, improving scalability and generalization over existing PINN methods.

Findings

Outperforms PINN on nonlinear PDEs in accuracy and scalability

Works effectively with irregular meshes and complex boundary conditions

Models trained on small domains generalize well to larger, complex 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.