Global error estimators for parametric monotone nonlinearities and neural approximations

2026-10-05

AI Summary

This study addresses the absence of globally reliable and computable neural network loss functions for parametric monotone nonlinear partial differential equations. To overcome this limitation, we propose error estimators based on operator splitting and discrete dual norms to construct variationally correct training objectives. Methodologically, the approach integrates first-order system least squares, discontinuous Petrov–Galerkin formulations, and computable surrogate dual norm techniques. Theoretically, monotonicity is leveraged to establish the two-sided reliability of the estimators over global trial spaces, thereby transcending the constraints of conventional local error estimation. This work achieves strict computability for arbitrary inputs, providing rigorous theoretical guarantees and an efficient training paradigm for neural network approximations of parameter-to-solution maps.

Abstract

We construct computable error estimators, which double as loss functions for neural networks, for a class of parametric nonlinear partial differential equations with a monotonicity property, and prove that they are globally reliable and efficient. The value of such a loss function is bounded above and below by the squared error in the natural trial norm, for every trial function, not merely for those near the exact solution; this global property rests on monotonicity. The construction rests on splitting the nonlinear operator into a linear part and a strongly monotone closure, and on measuring the linear residual in a discrete dual norm. Since the closure contributes a dual norm that admits no closed form when the trial norm is stronger than an $L_2$ norm, the estimator is built around a computable surrogate for it, required only to satisfy a pairing bound and a Lipschitz bound. The main theorem then yields two-sided bounds with explicit constants and covers two instances. A first-order system least-squares estimator on conforming trial spaces is the first instance studied: its two-sided bound holds on the whole trial space and therefore applies to arbitrary approximations, including those not in a discrete finite element space. The second instance is a discontinuous Petrov-Galerkin estimator on trial spaces of finite element functions for each parameter value, built with broken test spaces, for which no conformity is required and the dual norm is computed by independent element-local problems. All assumptions are verified for a model class of nonlinear fluxes, with the constants tracked explicitly in terms of the parameter range. Being computable for an arbitrary input, both estimators serve as variationally correct loss functions for neural network approximations of parameter-to-solution maps.

Problem

• parametric nonlinear PDEs
• global error estimators
• monotone nonlinearities
• neural network loss functions
• parameter-to-solution maps

Innovation

• computable error estimators
• monotone nonlinear PDEs
• neural network loss functions
• first-order system least-squares
• discontinuous Petrov-Galerkin