Residual-Based A Posteriori Error Bounds for Physics-Informed Neural Networks in Energy Seminorm

Physics-informed neural networks (PINNs) provide a flexible framework for solving partial differential equations, yet their training loss does not reliably reflect approximation error in Sobolev norms. In particular, PINNs may exhibit small training loss while incurring substantial error in the energy seminorm. This work develops a fully computable residual-based a posteriori error estimator for PINN approximations of coercive elliptic problems with soft Dirichlet boundary enforcement. The estimator augments the classical residual term with a harmonic lifting that accounts for boundary mismatch, yielding a reliable upper bound for the energy seminorm error. Reliability is established for constant-coefficient diffusion problems. For variable-coefficient diffusion, a Laplace-based lifting is used as a computable surrogate that remains practically reliable and consistent with the computational framework. A fully reproducible numerical study demonstrates effectivity indices between 1.05 and 1.31 across smooth, heterogeneous, and singular benchmark problems, including an L-shaped domain with a re-entrant corner. The results show that boundary mismatch is the dominant source of error, while residual-only estimators can underestimate the true error by orders of magnitude. These findings establish a principled and practical framework for certifying PINN solutions in the energy seminorm and highlight the necessity of incorporating boundary contributions for reliable error estimation.