  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.
