Learning time-dependent PDE solver using Message Passing Graph Neural Networks
This work addresses the problem of efficient PDE solving for computational physics and engineering, presenting an incremental advancement by adapting existing graph neural network methods to this domain.
The paper tackles the challenge of developing computationally efficient and accurate solvers for time-dependent partial differential equations by introducing a graph neural network approach using message-passing models, resulting in solvers that are independent of initial trained geometry and can handle complex domains.
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 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 trained geometry, i.e. the trained solver can find PDE solution on different complex domains. Lastly, we show that a recurrent graph neural network approach can find a temporal sequence of solutions to a PDE.