A One-Sided Level-Crossing Budget for Physics-Informed Neural Networks: Validated Mechanism, Absent Pathology

Physics-informed neural networks (PINNs) on stiff and unstable PDEs are widely reported to failthrough runaway high-frequency oscillation—most pointedly on the complex Ginzburg–Landau equation (CGLE), where a recent architecture-level fix (ASPEN) is premised on a conventional MLP PINN“catastrophically diverging into non-physical oscillations.” We design the natural loss-level counterpart:a one-sided level-crossing budget, built on the Kac–Rice/co-area machinery of our companion work, thatpenalizes the network field’s spatial crossing density only where it exceeds the physical budget of thesolution class. The loss is mesh-free, FFT-free, evaluated at the collocation points a PINN alreadyuses, costs one spatial derivative the residual already computes, and is provably inactive—zero loss, zerogradient—on fields within budget. We validate the mechanism in vitro: on sparse-data interpolationwhere a periodic-activation network demonstrably over-oscillates, the budget clamps the crossing profileto specification and improves test error on every seed. We then report, with full receipts, a negativeresult in vivo: across a faithful reimplementation of the ASPEN benchmark (two input conventions, upto the full published training depth) and a Benjamin–Feir-unstable chaotic testbed, the vanilla PINNalways fails—but always by under-resolving or decorrelating, never by excess oscillation. Its crossingdensity sits below the physical budget at every level, so a one-sided budget is verifiably inert, and uniform gradient damping is inert or harmful for the same reason. We conclude that loss-level oscillationsuppressors—ours included—address a failure mode whose prevalence is implementation-sensitive, andwe release a crossing-profile diagnostic that distinguishes over- from under-oscillatory PINN failure inone plot, together with all code, reference solvers, and raw runs.