H2(Ω) A Priori Error Bounds for Boundary-Adapted PINNs Solving Elliptic Dirichlet Problems
Abstract
We study the numerical resolution of uniformly elliptic Dirichlet boundary value problems (BVPs) by minimization of the PDE residual over a hypothesis space of Neural Networks made to satisfy exactly the boundary conditions through multiplication by a user-defined \emph{distance-to-boundary approximation} ρ. While an L2(∂Ω) boundary penalty cannot generally guarantee H2(Ω) error control, we show that exact enforcement can, provided ρ is chosen appropriately. More precisely, we identify a sharp sufficient condition on ρ preserving the regularity needed for H2(Ω) error control, provide practical constructions of such ρ for general smooth domains Ω, and refer to the resulting networks as \emph{boundary-adapted}. For boundary-adapted Rectified Quadratic Unit (ReQU) and hyperbolic tangent (tanh) networks, we then derive explicit high-probability H2(Ω) convergence rates by combining approximation-theoretic and statistical-learning arguments, making explicit the dependence on ρ. Numerical experiments support our theoretical results, showing that appropriate choices of ρ improve accuracy and convergence, while poorly chosen distance functions can substantially degrade the solution. Our proof also yields new VC-dimension bounds for hypothesis spaces of higher-order derivatives of ReQU and tanh networks, together with new approximation bounds for shallow ReQU networks in higher-order Sobolev norms, all of which are of important independent interest.
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BibTeX
@misc{tepakbong2026h2ω,
title = {{$H^2(Ω)$ A Priori Error Bounds for Boundary-Adapted PINNs Solving Elliptic Dirichlet Problems}},
author = {Nathanael Tepakbong and Jun Fan and Xiang Zhou and Ding-Xuan Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.19167},
eprint = {2607.19167},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.19167},
}
RIS
TY - UNPB
TI - H^2(Ω) A Priori Error Bounds for Boundary-Adapted PINNs Solving Elliptic Dirichlet Problems
AU - Nathanael Tepakbong
AU - Jun Fan
AU - Xiang Zhou
AU - Ding-Xuan Zhou
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2607.19167
UR - https://arxiv.org/abs/2607.19167
AB - We study the numerical resolution of uniformly elliptic Dirichlet boundary value problems (BVPs) by minimization of the PDE residual over a hypothesis space of Neural Networks made to satisfy exactly the boundary conditions through multiplication by a user-defined \emph{distance-to-boundary approximation} ρ. While an L^2(∂Ω) boundary penalty cannot generally guarantee H^2(Ω) error control, we show that exact enforcement can, provided ρ is chosen appropriately. More precisely, we identify a sharp sufficient condition on ρ preserving the regularity needed for H^2(Ω) error control, provide practical constructions of such ρ for general smooth domains Ω, and refer to the resulting networks as \emph{boundary-adapted}. For boundary-adapted Rectified Quadratic Unit (ReQU) and hyperbolic tangent (tanh) networks, we then derive explicit high-probability H^2(Ω) convergence rates by combining approximation-theoretic and statistical-learning arguments, making explicit the dependence on ρ. Numerical experiments support our theoretical results, showing that appropriate choices of ρ improve accuracy and convergence, while poorly chosen distance functions can substantially degrade the solution. Our proof also yields new VC-dimension bounds for hypothesis spaces of higher-order derivatives of ReQU and tanh networks, together with new approximation bounds for shallow ReQU networks in higher-order Sobolev norms, all of which are of important independent interest.
ER -