新闻 · arXiv cs.LG

A Statistical Formulation Gap for Nonlinear Multiscale Physics-Informed Learning

We prove a finite-sample formulation gap for physics-informed learning of nonlinear multiscale elliptic equations. For a uniformly monotone divergence-form class with coefficients oscillating at scale $ε$, we derive a finite-width, finite-sample, and finite-iteration error bound for a boundary-compatible variational neural solver. Its stability, sampling, and optimization constants are independent of $ε$, while all unresolved scale dependence is isolated in the best-approximation error. We then…