Learning Only On Boundaries: a Physics-Informed Neural operator for Solving Parametric Partial Differential Equations in Complex Geometries
This addresses the challenge of high computational cost and domain limitations for researchers and practitioners in computational physics and engineering, offering a novel approach with significant efficiency gains.
The paper tackles the problem of solving parametric partial differential equations (PDEs) in complex geometries by introducing a physics-informed neural operator that reduces training data requirements and handles unbounded domains, achieving a reduction in sample points from O(N^d) to O(N^{d-1}) and enabling solutions for previously unattainable unbounded problems.
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.