Neural PDE Solvers for Irregular Domains

TL;DR

This paper introduces a neural network framework capable of solving PDEs on irregular domains with complex boundaries, generalizing to unseen shapes and accurately imposing boundary conditions.

Contribution

It presents a novel approach for neural PDE solvers that handle irregular geometries and boundary conditions, with a differentiable method for domain interior-exterior classification.

Findings

Effective in solving PDEs on irregular domains

Generalizes well to unseen geometries

Provides theoretical error analysis

Abstract

Neural network-based approaches for solving partial differential equations (PDEs) have recently received special attention. However, the large majority of neural PDE solvers only apply to rectilinear domains, and do not systematically address the imposition of Dirichlet/Neumann boundary conditions over irregular domain boundaries. In this paper, we present a framework to neurally solve partial differential equations over domains with irregularly shaped (non-rectilinear) geometric boundaries. Our network takes in the shape of the domain as an input (represented using an unstructured point cloud, or any other parametric representation such as Non-Uniform Rational B-Splines) and is able to generalize to novel (unseen) irregular domains; the key technical ingredient to realizing this model is a novel approach for identifying the interior and exterior of the computational grid in a differentiable manner. We also perform a careful error analysis which reveals theoretical insights into several sources of error incurred in the model-building process. Finally, we showcase a wide variety of applications, along with favorable comparisons with ground truth solutions.