PhyGNNet: Solving spatiotemporal PDEs with Physics-informed Graph Neural Network

TL;DR

PhyGNNet introduces a graph neural network-based approach for solving PDEs, demonstrating improved fitting and extrapolation capabilities over traditional PINN models through experiments on Burgers and heat equations.

Contribution

This paper presents PhyGNNet, a novel GNN-based framework for PDEs that enhances fitting and extrapolation abilities compared to PINN models.

Findings

Outperforms PINN in fitting PDE solutions.

Shows better extrapolation in time and space.

Validated on Burgers and heat equations.

Abstract

Solving partial differential equations (PDEs) is an important research means in the fields of physics, biology, and chemistry. As an approximate alternative to numerical methods, PINN has received extensive attention and played an important role in many fields. However, PINN uses a fully connected network as its model, which has limited fitting ability and limited extrapolation ability in both time and space. In this paper, we propose PhyGNNet for solving partial differential equations on the basics of a graph neural network which consists of encoder, processer, and decoder blocks. In particular, we divide the computing area into regular grids, define partial differential operators on the grids, then construct pde loss for the network to optimize to build PhyGNNet model. What's more, we conduct comparative experiments on Burgers equation and heat equation to validate our approach, the results show that our method has better fit ability and extrapolation ability both in time and spatial areas compared with PINN.