Learning Only on Boundaries: A Physics-Informed Neural Operator for Solving Parametric Partial Differential Equations in Complex Geometries

作者：Zhiwei Fang, Sifan Wang, Paris Perdikaris · 发表于：Neural Computation · 年份：2024 · DOI：10.1162/neco_a_01647 · 被引用次数：17 · 研究领域：Model Reduction and Neural Networks、Numerical methods for differential equations、Numerical Methods and Algorithms

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 parameterized 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(Nd) to O(Nd-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 parameterized complex geometries and unbounded problems.