BOUNDARY-ADAPTED PINNS FOR ELLIPTIC DIRICHLET PROBLEMS: H2(Ω) A PRIORI ERROR BOUNDS WITH APPLICATION TO MEAN ESCAPE TIME COMPUTATION

This paper investigates the theoretical foundations of Physics-Informed Neural Networks (PINNs) when applied to elliptic Dirichlet boundary value problems, specifically motivated by the computation of Mean Escape Time (MET) for stochastic processes. The authors examine the limitations of standard PINN formulations—where boundary conditions are often enforced via soft penalties—and propose a rigorous framework for "boundary-adapted" PINNs. These networks enforce Dirichlet conditions exactly by multiplying the neural network output with a predefined distance-to-boundary function, ρ.

To establish theoretical guarantees, the authors employ approximation theory and statistical learning theory. They derive a priori error bounds in the H2(Ω) Sobolev norm, which is critical for problems like MET that exhibit sharp boundary layers. The analysis covers both Rectified Quadratic Unit (ReQU) and hyperbolic tangent (tanh) activation functions.

The study reveals that exact boundary enforcement is a necessary but insufficient condition for achieving H2(Ω) error bounds. The authors demonstrate that the distance function ρ must be a smooth approximation that is normalized to the first order. When this condition is met, the boundary-adapted PINN architecture provides near-minimax optimal convergence rates relative to the number of collocation points.

Additionally, the paper provides several contributions of independent interest:

• New W2,∞(Ω) approximation error bounds for shallow ReQU networks.
• Novel VC-dimension bounds for higher-order derivatives of ReQU and tanh networks, which allow for error control without requiring restrictive weight constraints.
• Numerical evidence confirming that the choice of ρ significantly impacts accuracy, with poorly chosen distance functions leading to substantial performance degradation.

Numerical computation of MET is notoriously difficult due to the stochastic nature of the underlying processes and the presence of sharp boundary layers. Traditional methods often struggle with high-dimensional domains or require extensive domain-specific knowledge. By providing the first rigorous H2(Ω) error analysis for boundary-enforced PINNs, this work offers a principled guide for practitioners to design more stable and accurate neural network architectures for solving elliptic PDEs, moving beyond heuristic approaches to boundary condition enforcement.