2026/07/21 by Nathanael Tepakbong, Jun Fan, Xiang Zhou +1
#math.NA #cs.LG #cs.NA #math.ST #stat.ML #stat.TH
Motivated by the numerical computation of the Mean Escape Time (MET) τ:Ω→ℝ of a stochastic process from a bounded domain Ω⊆ℝd, 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 H2(Ω) error bounds, and that a sufficient and essentially necessary condition is for ρ 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 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.