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

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 $\varepsilon$. Assuming a quantitative corrected $H^1$-estimate, a two-scale state class yields

$$\mathcal A_m^\varepsilon \le C\bigl(\varepsilon+Φ_{0,m_0}^2+Φ_{1,m_1}^2\bigr)$$

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 $O(\varepsilon)$ 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 $(\varepsilon\sqrt N)^{-1}$ and $(\varepsilon^2\sqrt N)^{-1}$, respectively. These optimizer-independent lower bounds hold in every spatial dimension. Numerical experiments confirm the predicted $\varepsilon$- 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 $H^1$ errors as the microscopic scale is refined.