Inverted model selection in physics-informed neural networks: when a lower residual selects a worse solution

Abstract

Physics-informed neural networks (PINNs) are commonly evaluated via a single aggregate residual, assuming a smaller residual indicates a better solution. Testing this directly across three constrained PDE systems, I find this assumption can systematically fail. In matched pairs of solvers differing only in whether a defining structural identity is hard-wired or penalized, the penalized variant frequently attains a lower equation residual while violating that identity by several orders of magnitude, causing the exact variant to be falsely ranked worse. Over 64 matched pairs spanning two systems, four network variants, and eight seeds, this inversion occurs in 83\% of cases (95\% Wilson CI: 72--90\%), with rates from 72\% to 94\% across systems. Testing across six architectures--MLP, cPINN, XPINN, hp-VPINN, and physics-informed DeepONet and FNO--inverts the ranking in 46 of 48 pairs, indicating that this variability is problem-dependent rather than specific to the approximator. A third, larger vorticity--streamfunction problem shows the same ordering: the residual-optimal solver violates its structural identity by over six orders above tolerance, despite a residual margin of only 9.37%. Because a scalar loss cannot expose this, I introduce a lexicographic admissibility gate spanning the structural identity, boundary trace, and a solvability integral that must vanish independently of the equation residual. All three must pass before residuals can compete. This gate catches three artifact classes but misses a fourth: a prescribed-structure prior yields fields that pass every single-run check, yet deleting the source term reveals that 97\% of the reported structure survives removal of the physics. Reference data and figures accompany the paper.