Learning Only On Boundaries: a Physics-Informed Neural operator for Solving Parametric Partial Differential Equations in Complex Geometries

TL;DR

This paper introduces a physics-informed neural operator that efficiently solves parametrized PDEs in complex and unbounded geometries by training solely on boundary data, significantly reducing sample requirements and extending applicability.

Contribution

The method reformulates PDEs into boundary integral equations, enabling boundary-only training and handling unbounded problems, which improves efficiency and scope over existing neural operators.

Findings

Reduces training sample complexity from O(N^d) to O(N^{d-1})

Successfully solves complex geometries and unbounded PDE problems

Outperforms traditional PINNs in efficiency and applicability

Abstract

Recently deep learning surrogates and neural operators have shown promise in solving partial differential equations (PDEs). However, they often require a large amount of training data and are limited to bounded domains. In this work, we present a novel physics-informed neural operator method to solve parametrized boundary value problems without labeled data. By reformulating the PDEs into boundary integral equations (BIEs), we can train the operator network solely on the boundary of the domain. This approach reduces the number of required sample points from $O(N^d)$ to $O(N^{d-1})$, where $d$ is the domain's dimension, leading to a significant acceleration of the training process. Additionally, our method can handle unbounded problems, which are unattainable for existing physics-informed neural networks (PINNs) and neural operators. Our numerical experiments show the effectiveness of parametrized complex geometries and unbounded problems.