2026/07/17 by Ronald Katende
#math.NA #cs.LG #cs.NA #math.AP
We develop a non-asymptotic approximation, sampling, and finite-iteration optimization theory for variational physics-informed approximation of uniformly monotone nonlinear multiscale elliptic equations. For boundary-compatible neural feature classes, the population error splits into approximation, empirical quadrature, and projected-gradient terms, with all non-approximation constants uniform in the microscopic scale \(ε\). Assuming a quantitative corrected \(H1\)-estimate, a two-scale state class yields \mathcal Amε ≤ C(ε+Φ0,m02+Φ1,m12) in arbitrary dimension. We further introduce a convex primal-dual physics loss whose population value is a computable upper certificate for the state error. With additional flux-corrector regularity, a divergence-compatible two-scale flux class gives a certified state-flux bound combining \(O(ε)\) approximation, state and flux feature errors, empirical sampling error, and an \(O(K-1)\) optimization term. In contrast, for general periodic nonlinear fluxes satisfying a natural nondegeneracy condition, the empirical Rademacher complexities of strong-residual and squared-residual classes are bounded below by constant multiples of \((ε√ N)-1\) and \((ε2√ N)-1\), respectively. These optimizer-independent lower bounds hold in every spatial dimension. Numerical experiments confirm the predicted \(ε\)- and \(N\)-scalings for nonlinear fluxes in \(d=1,2,3\), validate every computed primal-dual certificate, and show that corrector-enriched classes substantially reduce energy and \(H1\) errors as the microscopic scale is refined.