Variationally correct loss functions for nonlinear parametric problems and their neural approximations
Year: 2026
Type: preprint
Abstract: We construct computable a posteriori error estimators for parametric families of nonlinear elliptic problems by measuring the residual of a variational formulation in the dual norm of a closed subspace of the test space. A single abstract theorem shows that the resulting estimator is bounded above and below by the error in the trial norm, i.e., it is a variationally correct loss function. Two instances are covered. The first considers an $L_2$-like test space, yielding a nonlinear first-order system least-squares loss function that certifies arbitrary, possibly meshless, approximations. The second breaks the test space, yielding a nonlinear discontinuous Petrov-Galerkin loss function computed by element-local problems. All assumptions are verified for quasilinear diffusion, including monotone fluxes and semilinear problems, and a nonlocal Kirchhoff model. Numerical studies train neural networks approximating the parameter-to-solution maps with the latter loss and measure the ratio of loss to squared error.
Source: arXiv (Cornell University)
Institutions: Portland State University, University of South Carolina
Language: English
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: green
Awards: FRG: Collaborative Research: Variationally Stable Neural Networks for Simulation, Learning, and Experimental Design of Complex Physical Systems, Accuracy Controlled Optimization in Physics Informed Deep Learning, G620044194, FRG: Collaborative Research: Variationally Stable Neural Networks for Simulation, Learning, and Experimental Design of Complex Physical Systems, RTG: Mathematical Foundation of Data Science at University of South Carolina
OpenAlex ID: W7222177003