Neural functional a posteriori error estimates

TL;DR

This work introduces a loss function for supervised and physics-informed neural network training that embeds functional a posteriori error estimates to obtain rigorous error majorants. By learning auxiliary physical fields, the networks produce certified error bounds via a cheap postprocessing stage, applicable to elliptic PDEs and various neural operators (FNO, fSNO, ChebNO, DilResNet, UNet, MLP). Empirical results in 1D and 2D demonstrate competitive accuracy across architectures and show that the associated upper bounds on error track true error while remaining inexpensive to compute. Overall, the approach provides a framework for simultaneous predictive accuracy and certified error control in neural PDE solvers, bridging physics-informed learning with rigorous a posteriori error guarantees.

Abstract

We propose a new loss function for supervised and physics-informed training of neural networks and operators that incorporates a posteriori error estimate. More specifically, during the training stage, the neural network learns additional physical fields that lead to rigorous error majorants after a computationally cheap postprocessing stage. Theoretical results are based upon the theory of functional a posteriori error estimates, which allows for the systematic construction of such loss functions for a diverse class of practically relevant partial differential equations. From the numerical side, we demonstrate on a series of elliptic problems that for a variety of architectures and approaches (physics-informed neural networks, physics-informed neural operators, neural operators, and classical architectures in the regression and physics-informed settings), we can reach better or comparable accuracy and in addition to that cheaply recover high-quality upper bounds on the error after training.