Neural functional a posteriori error estimates

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.