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

2026-07-21

AI Summary

This work addresses the challenge of achieving high-order Sobolev accuracy in physics-informed neural networks (PINNs) for elliptic Dirichlet boundary value problems. The authors propose boundary-adaptive PINNs that exactly embed Dirichlet conditions into the network output by multiplying with a first-order normalized smooth boundary distance function ρ. Through a combined analysis grounded in approximation theory and statistical learning, they establish—for the first time—that merely satisfying boundary conditions is insufficient to guarantee an $H^2(\Omega)$ error bound, and rigorously identify the necessity of the first-order normalization property for ρ. Leveraging ReQU or tanh activation functions, they derive novel VC-dimension bounds for derivative hypothesis spaces and high-order Sobolev approximation rates for shallow networks, leading to a provable $H^2$ priori error estimate. Numerical experiments confirm that a properly constructed ρ significantly enhances both accuracy and convergence in MET computations.

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.

Problem

• Elliptic Dirichlet Problems
• Mean Escape Time
• Physics-Informed Neural Networks
• A Priori Error Bounds
• Boundary Enforcement

Innovation

• Boundary-Adapted PINNs
• H2 error bounds
• exact boundary enforcement
• smooth distance function
• ReQU networks