Variationally correct loss functions for nonlinear parametric problems and their neural approximations
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 L2-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.
Related
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BibTeX
@misc{castillo2026variationally,
title = {{Variationally correct loss functions for nonlinear parametric problems and their neural approximations}},
author = {Pablo Cortés Castillo and Wolfgang Dahmen and Jay Gopalakrishnan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19296},
eprint = {2610.19296},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19296},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Variationally correct loss functions for nonlinear parametric problems and their neural approximations
AU - Pablo Cortés Castillo
AU - Wolfgang Dahmen
AU - Jay Gopalakrishnan
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.19296
UR - https://arxiv.org/abs/2610.19296
AB - 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.
ER -