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

arXiv: https://arxiv.org/abs/2607.15702 · HTML: https://mlsanity.com/paper/2607.15702/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 12 references that are in the map.
- First posted on arXiv 17 Jul 2026.

## Builds on (most cited first, 4 of 4)

- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Variational Physics-Informed Neural Networks For Solving Partial Differential Equations](https://mlsanity.com/paper/1912.00873/) (2019)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs](https://mlsanity.com/paper/2006.16144/) (2020)
- [First Order System Least Squares Neural Networks](https://mlsanity.com/paper/2409.20264/) (2024)

## Abstract (arXiv)

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 (H1)-estimate, a two-scale state class yields [ mathcal Am^varepsilon ≤ Cbigl(varepsilon+Phi0,m02+Phi1,m12bigr) ] 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√ N)-1) and ((varepsilon2√ 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 (H1) errors as the microscopic scale is refined.