Learning Time-dependent PDE Solver using Message Passing Graph Neural Networks

Pourya Pilva, Ahmad Zareei

Keywords: graph neural networks, partial differential equations, time-dependent PDE, message passing graph neural networks

Abstract: One of the main challenges in solving time-dependent partial differential equations is to develop computationally efficient solvers that are accurate and stable. Here, we introduce a general graph neural network approach to finding efficient PDE solvers through learning using message-passing models. We first introduce domain invariant features for PDE-data inspired by classical PDE solvers for an efficient physical representation. Next, we use graphs to represent PDE-data on an unstructured mesh and show that message passing graph neural networks (MPGNN) can parameterize governing equations, and as a result, efficiently learn accurate solver schemes for linear/nonlinear PDEs. We further show that the solvers are independent of the initial training geometry and can solve the same PDE on more complex domains. Lastly, we show that a recurrent graph neural network approach can find a temporal sequence of solutions to a PDE.

One-sentence Summary: Solving Time-dependent PDEs using (Recurrent) Message Passing Graph Neural Networks