An Implicit GNN Solver for Poisson-like problems

TL;DR

The paper introduces $ ext{ extbackslash Psi}$-GNN, a physics-informed implicit GNN approach for solving Poisson PDEs with mixed boundary conditions, capable of handling unstructured meshes and providing convergence guarantees.

Contribution

It presents the first implicit, physics-informed GNN that models infinitely deep networks, explicitly incorporates boundary conditions, and guarantees convergence for Poisson problems.

Findings

Accurately solves Poisson problems on unstructured meshes.

Handles various boundary conditions with a single trained model.

Provides theoretical convergence guarantees.

Abstract

This paper presents $\Psi$-GNN, a novel Graph Neural Network (GNN) approach for solving the ubiquitous Poisson PDE problems with mixed boundary conditions. By leveraging the Implicit Layer Theory, $\Psi$-GNN models an "infinitely" deep network, thus avoiding the empirical tuning of the number of required Message Passing layers to attain the solution. Its original architecture explicitly takes into account the boundary conditions, a critical prerequisite for physical applications, and is able to adapt to any initially provided solution. $\Psi$-GNN is trained using a "physics-informed" loss, and the training process is stable by design, and insensitive to its initialization. Furthermore, the consistency of the approach is theoretically proven, and its flexibility and generalization efficiency are experimentally demonstrated: the same learned model can accurately handle unstructured meshes of various sizes, as well as different boundary conditions. To the best of our knowledge, $\Psi$-GNN is the first physics-informed GNN-based method that can handle various unstructured domains, boundary conditions and initial solutions while also providing convergence guarantees.