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

Abstract: Motivated by the numerical computation of the Mean Escape Time (MET) $\tau:\Omega\to\mathbb{R}$ of a stochastic process from a bounded domain $\Omega\subseteq\mathbb{R}^d$, 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 $\rho$. 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 $\rho$. In particular, we show that exact boundary enforcement alone is not enough for $H^2(\Omega)$ error bounds, and that a sufficient and essentially necessary condition is for $\rho$ to be a smooth distance approximation *normalized to first order*, of the kind constructed in arXiv:2104.08426 [math.NA]. We thereby identify this subclass of *boundary-adapted* PINNs as the appropriate neural network ansatz for solving Dirichlet BVPs. Numerical experiments support the theory, 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.

Numerical Analysis (math.NA); Machine Learning (cs.LG); Statistics Theory (math.ST); Machine Learning (stat.ML)

arXiv:2607.19167 [math.NA]

(or arXiv:2607.19167v1 [math.NA] for this version)

https://doi.org/10.48550/arXiv.2607.19167 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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