SSBE-PINN: A Sobolev Boundary Scheme Boosting Stability and Accuracy in Elliptic/Parabolic PDE Learning

TL;DR

This paper introduces SSBE-PINN, a Sobolev boundary scheme that enhances the stability and accuracy of physics-informed neural networks in solving elliptic and parabolic PDEs by incorporating boundary regularity through Sobolev norms.

Contribution

The paper proposes a novel Sobolev-Stable Boundary Enforcement method that improves boundary treatment in PINNs, providing theoretical stability guarantees and demonstrating superior numerical performance.

Findings

SSBE-PINN achieves lower H1 and L2 errors compared to standard PINNs.

The method maintains stability and accuracy in high-dimensional PDE problems.

Numerical experiments validate the theoretical stability and robustness of SSBE-PINN.

Abstract

Physics-Informed Neural Networks (PINNs) have emerged as a powerful framework for solving partial differential equations (PDEs), yet they often fail to achieve accurate convergence in the H1 norm, especially in the presence of boundary approximation errors. In this work, we propose a novel method called Sobolev-Stable Boundary Enforcement (SSBE), which redefines the boundary loss using Sobolev norms to incorporate boundary regularity directly into the training process. We provide rigorous theoretical analysis demonstrating that SSBE ensures bounded H1 error via a stability guarantee and derive generalization bounds that characterize its robustness under finite-sample regimes. Extensive numerical experiments on linear and nonlinear PDEs, including Poisson, heat, and elliptic problems, show that SSBE consistently outperforms standard PINNs in terms of both relative L2 and H1 errors, even in high-dimensional settings. The proposed approach offers a principled and practical solution for improving gradient fidelity and overall solution accuracy in neural network based PDE solvers.