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

Abstract

The great success of Physics-Informed Neural Network (PINN) in addressing partial differential equations (PDEs) has enhanced our ability to simulate and understand complex physical systems across various science and engineering disciplines. Despite their achievements, existing PINN-like methods often face limitations in scalability and are primarily effective within in-sample scenarios. To overcome these challenges, this work proposes a novel discrete approach termed Physics-Informed Graph Neural Network (PIGNN) to solve both forward and inverse problems associated with nonlinear PDEs. Our approach seamlessly integrates the strength of graph neural network (GNN), physical laws and finite difference method to approximate the solutions of physical systems. Through a series of comprehensive numerical experiments, we compare the performance of our PIGNN against the established PINN baseline using three well-known nonlinear PDEs: the heat equation, the Burgers equation, and the FitzHugh-Nagumo equation. Experimental outcomes highlight the superior performance of our PIGNN in handling irregular meshes, long time steps, flexible spatial resolutions, and diverse initial and boundary conditions. These results also demonstrate the superiority of our approach in terms of accuracy, time extrapolability, generalizability and scalability. A key advantage of our approach lies in its exceptional adaptability: models initially trained on small, simplified domains exhibit robust fitting capabilities that can be seamlessly transferred to more complex, larger-scale scenarios.