arXiv:2609.01557 2026 Training 2 ideas extracted · analyzed Sep 2, 2026
The paper provides a constructive zero-temperature variational principle for selecting weak KAM subsolutions: minimizing an exponential integral asymptotically minimizes the worst-case Hamilton–Jacobi violation. Its transferable asset is the combination of convex Hamiltonian structure, a globally normalized potential, and a theorem identifying a dynamically critical contact set. In neural PDE solvers and learned dynamical models, this suggests replacing average residual fitting with an annealed soft supremum and using long-time Lax–Oleinik near-contact scores for adaptive sampling. The approach is especially suitable for periodic value functions where worst-case constraint satisfaction and localization of invariant critical regions matter.
Use the discrepancy between a learned potential and its long-time backward Lax–Oleinik evolution to identify dynamically critical states. Persistent near-contact points are candidates for the Aubry set and can guide adaptive collocation, while states with large gaps can receive fewer training samples.
Train a neural periodic potential to minimize an exponential variational functional rather than a mean-squared Hamilton–Jacobi residual. Increasing the inverse-temperature parameter concentrates optimization on the worst violating locations, encouraging a learned critical subsolution whose equality set represents dynamically important regions.