Boundary-Adapted PINNs for Elliptic Dirichlet Problems: $H^2(Ω)$ A Priori Error Bounds with Application to Mean Escape Time Computation

• URL: http://arxiv.org/abs/2607.19167v1
• Date: Tue, 21 Jul 2026 15:03:42 GMT
• Title: Boundary-Adapted PINNs for Elliptic Dirichlet Problems: $H^2(Ω)$ A Priori Error Bounds with Application to Mean Escape Time Computation
• Abstract summary: We study elliptic Dirichlet boundary value problems (BVPs) using boundary-enforced Physics-Informed Neural Networks (PINNs)<n>We derive a priori error bounds that make explicit the dependence on $$.<n>Our proof also yields new VC-dimension bounds for hypothesis spaces of higher-order derivatives of ReQU and tanh networks.
• Score: 14.109379170744546
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Motivated by the numerical computation of the Mean Escape Time (MET) $τ:Ω\to\mathbb{R}$ of a stochastic process from a bounded domain $Ω\subseteq\mathbb{R}^d$, we study elliptic Dirichlet boundary value problems (BVPs) using boundary-enforced Physics-Informed Neural Networks (PINNs), in which the Dirichlet condition is imposed exactly by multiplying the network output with a predefined distance-to-boundary approximation $ρ$. Combining approximation-theoretic and statistical-learning arguments for Rectified Quadratic Unit (ReQU) and hyperbolic tangent (tanh) networks, we derive a priori error bounds that make explicit the dependence on $ρ$. In particular, we show that exact boundary enforcement alone is not enough for $H^2(Ω)$ error bounds, and that a sufficient and essentially necessary condition is for $ρ$ to be a smooth distance approximation $\textit{normalized to first order}$, of the kind constructed in arXiv:2104.08426 [math.NA]. We thereby identify this subclass of $\textit{boundary-adapted}$ PINNs as the appropriate neural network ansatz for solving Dirichlet BVPs. Numerical experiments support the theory, 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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