SSBE-PINN: A Sobolev Boundary Scheme Boosting Stability and Accuracy in Elliptic/Parabolic PDE Learning
Qixuan Zhou, Chuqi Chen, Tao Luo, Yang Xiang
TL;DR
This work tackles the challenge of achieving robust $H^1$ convergence in neural PDE solvers by identifying weakness in standard PINN boundary losses. It introduces Sobolev-Stable Boundary Enforcement (SSBE), which replaces or augments the boundary term with an $H^1(\\partial\\Omega)$ penalty computed in intrinsic, boundary-local coordinates, and couples this with a boundary-flattening strategy to handle complex geometries. The authors establish $H^1$-stability findings, providing bounds of the form $\\|u-v\\|_{H^1(\\Omega)}^2 \le C \\mathcal{R}_{D}(v)$, and derive a priori and a posteriori generalization bounds based on Barron spaces and Rademacher complexity for both elliptic and parabolic PDEs. Extensive numerical experiments across Poisson, heat, nonlinear elliptic, high-dimensional, and irregular-domain problems demonstrate that SSBE consistently improves relative $L^2$ and $H^1$ errors over standard PINNs, highlighting its practical impact on gradient fidelity and solution accuracy in neural PDE solvers.
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.