$H^2(Ω)$ A Priori Error Bounds for Boundary-Adapted PINNs Solving Elliptic Dirichlet Problems

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 distance-to-boundary approximation ρ. While an L^2(\partialΩ) 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 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.