Incorporating Local H\"older Regularity into PINNs for Solving Elliptic PDEs

TL;DR

This paper enhances physics-informed neural networks for elliptic PDEs by integrating local H"older regularity, leading to improved accuracy and robustness through a novel loss function and sampling strategy.

Contribution

It introduces a new local H"older regularization term into PINNs and develops a variable-distance sampling method to improve PDE solution accuracy.

Findings

Significant accuracy improvements over standard PINNs

Enhanced robustness in elliptic PDE solutions

Effective error estimates for the proposed method

Abstract

In this paper, local H\"older regularization is incorporated into a physics-informed neural networks (PINNs) framework for solving elliptic partial differential equations (PDEs). Motivated by the interior regularity properties of linear elliptic PDEs, a modified loss function is constructed by introducing local H\"older regularization term. To approximate this term effectively, a variable-distance discrete sampling strategy is developed. Error estimates are established to assess the generalization performance of the proposed method. Numerical experiments on a range of elliptic problems demonstrate notable improvements in both prediction accuracy and robustness compared to standard physics-informed neural networks.