Non-Asymptotic Variational Learning for Monotone Nonlinear Multiscale Elliptic Equations: Scale-Robust Primal-Dual Bounds and Strong-Form Statistical Ill-Conditioning

What the math gives to ML

The paper identifies two transferable mechanisms for neural PDE solvers: multiscale feature enrichment that removes dependence on the microscopic scale from approximation error, and a convex primal-dual objective whose value certifies state error rather than serving only as a heuristic training loss. The key architectural lesson is to represent both the macroscopic field and oscillatory corrector explicitly instead of asking a standard network to resolve frequencies of order $1/\varepsilon$. The key optimization lesson is to train a state network together with a flux network and monitor a computable Fenchel-gap certificate. Its strongest immediate application is variational PINNs or neural operators for rapidly oscillatory elliptic problems, where strong-form residuals become statistically ill-conditioned as $\varepsilon$ decreases.