AI and the World's Hard Problems · Mathematics · arXiv · 2026-10-21
$H^2(\Omega)$ A Priori Error Bounds for Boundary-Adapted PINNs Solving Elliptic Dirichlet Problems
The study tackles the numerical solution of uniformly elliptic Dirichlet boundary value problems by minimizing the PDE residual over neural networks made to satisfy boundary conditions exactly through multiplication by a user-defined distance-to-boundary approximation ρ. The authors identify a sharp sufficient condition on ρ preserving the regularity needed for H²(Ω) error control, give practical constructions for general smooth domains, and derive explicit high-probability H²(Ω) convergence rates for boundary-adapted ReQU and tanh networks. Numerical experiments support the theory, showing that appropriate choices of ρ improve accuracy and convergence while poorly chosen distance functions substantially degrade the solution.