Boundary-Adapted PINNs for Elliptic Dirichlet Problems: $H^2(Ω)$ A Priori Error Bounds with Application to Mean Escape Time Computation
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 enforc...
| Authors | Nathanael Tepakbong, Jun Fan, Xiang Zhou et al. |
|---|---|
| Primary category | math.NA |
| Published | 21 July 2026 |
| Last updated at source | 21 July 2026 |
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