$H^2(\Omega )$ 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} $\rho$. While an $L^2(\partial \Omega )$ boundary penalty cannot generally guarantee $H^2(\Omega )$ error control, we show that exact enforcement can, provided $\rho$ is chosen appropriately. More precisely, we identify a sharp sufficient condition on $\rho$ preserving the regularity needed for $H^2(\Omega )$ error control, provide practical constructions of such $\rho$ for general smooth domains $\Omega$, 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 $H^2(\Omega )$ convergence rates by combining approximation-theoretic and statistical-learning arguments, making explicit the dependence on $\rho$. Numerical experiments support our theoretical results, showing that appropriate choices of $\rho$ 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.