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.

翻译：物理信息神经网络（PINN）在求解偏微分方程（PDEs）方面取得的巨大成功，显著推进了我们对科学与工程中复杂物理系统的模拟与理解。然而，许多类PINN方法可扩展性较差，且局限于样本内场景。为应对这些挑战，本研究提出一种新颖的离散化方法，称为物理信息图神经网络（PIGNN），用于求解正向与反向非线性偏微分方程。特别地，我们的方法无缝整合了图神经网络（GNN）、物理方程与有限差分法的优势，以逼近物理系统的解。我们在三个经典非线性偏微分方程（热传导方程、Burgers方程与FitzHugh-Nagumo方程）上将所提方法与PINN基线进行了对比。通过开展大量数值实验，我们证明了该方法在处理不规则网格、更长时步、任意空间分辨率、变化初始条件（ICs）与边界条件（BCs）方面具有优异性能。数值结果亦表明我们的方法在精度、时间外推性、泛化能力与可扩展性方面具有优越性。本方法的主要优势在于：在简单设置的小型区域上训练的模型具备出色的拟合能力，可直接应用于大型区域中更复杂的场景。