Extended Interface Physics-Informed Neural Networks Method for Moving Interface Problems

TL;DR

This paper introduces an extended PINN framework for moving interface problems, utilizing level set functions and NTK theory to improve accuracy and convergence, validated through numerical experiments on complex PDEs.

Contribution

The paper presents a novel XI-PINN method incorporating level set functions and NTK theory, advancing PINN capabilities for moving interface PDEs with rigorous error analysis.

Findings

Faster training convergence compared to traditional PINNs

Accurate approximation of moving interface problems

Validated robustness through numerical experiments

Abstract

Physics-Informed Neural Networks (PINNs) have emerged as a powerful class of mesh-free numerical methods for solving partial differential equations (PDEs), particularly those involving complex geometries. In this work, we present an innovative Extended Interface Physics-Informed Neural Network (XI-PINN) framework specifically designed to solve parabolic moving interface problems. The proposed approach incorporates a level set function to characterize the interface, which can be obtained either directly or through a neural network solution. We conduct a rigorous a priori error analysis for the XI-PINN method, providing error bounds for the approximation. Leveraging the Neural Tangent Kernel (NTK) theory, we further demonstrate that XI-PINN achieves a faster training convergence rate compared to conventional PINN approaches. The method's versatility is further demonstrated by its application to the Oseen equations. We perform comprehensive numerical experiments to validate the efficacy, accuracy, and robustness of the proposed framework.