Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problems

TL;DR

This paper introduces a graph neural network-based physics-informed framework that effectively solves complex forward and inverse PDE problems, overcoming scalability, boundary enforcement, and irregular geometry challenges of traditional PINNs.

Contribution

The paper proposes a novel discrete PINN framework using GCNs and PDE variational principles, enabling better scalability, boundary condition enforcement, and handling of irregular geometries.

Findings

Successfully applied to various linear and nonlinear PDEs.

Improved training efficiency and convergence over traditional PINNs.

Effective handling of irregular geometries with unstructured meshes.

Abstract

Despite the great promise of the physics-informed neural networks (PINNs) in solving forward and inverse problems, several technical challenges are present as roadblocks for more complex and realistic applications. First, most existing PINNs are based on point-wise formulation with fully-connected networks to learn continuous functions, which suffer from poor scalability and hard boundary enforcement. Second, the infinite search space over-complicates the non-convex optimization for network training. Third, although the convolutional neural network (CNN)-based discrete learning can significantly improve training efficiency, CNNs struggle to handle irregular geometries with unstructured meshes. To properly address these challenges, we present a novel discrete PINN framework based on graph convolutional network (GCN) and variational structure of PDE to solve forward and inverse partial differential equations (PDEs) in a unified manner. The use of a piecewise polynomial basis can reduce the dimension of search space and facilitate training and convergence. Without the need of tuning penalty parameters in classic PINNs, the proposed method can strictly impose boundary conditions and assimilate sparse data in both forward and inverse settings. The flexibility of GCNs is leveraged for irregular geometries with unstructured meshes. The effectiveness and merit of the proposed method are demonstrated over a variety of forward and inverse computational mechanics problems governed by both linear and nonlinear PDEs.