# Global error estimators for parametric monotone nonlinearities and neural approximations

arXiv: https://arxiv.org/abs/2610.05767 · HTML: https://mlsanity.com/paper/2610.05767/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 33 references that are in the map.
- First posted on arXiv 5 Oct 2026.

## Builds on (most cited first, 9 of 9)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Variational Physics-Informed Neural Networks For Solving Partial Differential Equations](https://mlsanity.com/paper/1912.00873/) (2019)
- [Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems](https://mlsanity.com/paper/1809.08327/) (2018)
- [Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEs](https://mlsanity.com/paper/2106.14473/) (2021)
- [Deep least-squares methods: an unsupervised learning-based numerical method for solving elliptic PDEs](https://mlsanity.com/paper/1911.02109/) (2019)
- [Data driven approximation of parametrized PDEs by Reduced Basis and Neural Networks](https://mlsanity.com/paper/1904.01514/) (2019)
- [Variationally correct operator learning: Reduced basis neural operator with a posteriori error estimation](https://mlsanity.com/paper/2512.21319/) (2025)
- [DPG loss functions for learning parameter-to-solution maps by neural networks](https://mlsanity.com/paper/2506.18773/) (2025)

## Abstract (arXiv)

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 L2 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.