Physics-Informed Tailored Finite Point Operator Network for Parametric Interface Problems

TL;DR

This paper introduces a physics-informed neural network method called PI-TFPONet for solving parametric interface problems in PDEs, avoiding labeled data and training difficulties, with proven convergence and superior performance.

Contribution

The novel PI-TFPONet method leverages physical prior knowledge, eliminating the need for PDE residuals in training, and is proven to converge for small mesh sizes and in singularly perturbed problems.

Findings

Achieves comparable or better accuracy than supervised methods

Proven to converge with small mesh sizes and minimized loss

Uniform convergence for singularly perturbed interface problems

Abstract

Learning operators for parametric partial differential equations (PDEs) using neural networks has gained significant attention in recent years. However, standard approaches like Deep Operator Networks (DeepONets) require extensive labeled data, and physics-informed DeepONets encounter training challenges. In this paper, we introduce a novel physics-informed tailored finite point operator network (PI-TFPONet) method to solve parametric interface problems without the need for labeled data. Our method fully leverages the prior physical information of the problem, eliminating the need to include the PDE residual in the loss function, thereby avoiding training challenges. The PI-TFPONet is specifically designed to address certain properties of the problem, allowing us to naturally obtain an approximate solution that closely matches the exact solution. Our method is theoretically proven to converge if the local mesh size is sufficiently small and the training loss is minimized. Notably, our approach is uniformly convergent for singularly perturbed interface problems. Extensive numerical studies show that our unsupervised PI-TFPONet is comparable to or outperforms existing state-of-the-art supervised deep operator networks in terms of accuracy and versatility.