Astral: training physics-informed neural networks with error majorants

TL;DR

Astral introduces a loss function for physics-informed neural networks based on functional a posteriori error majorants, providing a direct upper bound on the energy-norm error $E[\phi-\tilde{\phi}]$ rather than relying on PDE residuals. By deriving problem-specific majorants for diffusion, Maxwell, and convection–diffusion equations, the method enables simultaneous training and a posteriori error control, with a training objective that minimizes the majorant $U[\tilde{\phi},\mathcal{D},w]$. Across seven PDEs, Astral is generally competitive with residual losses and often yields faster convergence and lower errors, particularly in highly anisotropic or geometrically challenging settings (e.g., Maxwell with $\alpha=1$), while providing informative bounds on the true error. Limitations include the need to derive appropriate majorants for each PDE and potential numerical challenges in evaluating the bounds, but the approach offers a principled path to reliable error estimation in PiNNs.

Abstract

The primal approach to physics-informed learning is a residual minimization. We argue that residual is, at best, an indirect measure of the error of approximate solution and propose to train with error majorant instead. Since error majorant provides a direct upper bound on error, one can reliably estimate how close PiNN is to the exact solution and stop the optimization process when the desired accuracy is reached. We call loss function associated with error majorant $\textbf{Astral}$: neur$\textbf{A}$l a po$\textbf{ST}$erio$\textbf{RI}$ function$\textbf{A}$l Loss. To compare Astral and residual loss functions, we illustrate how error majorants can be derived for various PDEs and conduct experiments with diffusion equations (including anisotropic and in the L-shaped domain), convection-diffusion equation, temporal discretization of Maxwell's equation, and magnetostatics problem. The results indicate that Astral loss is competitive to the residual loss, typically leading to faster convergence and lower error (e.g., for Maxwell's equations, we observe an order of magnitude better relative error and training time). We also report that the error estimate obtained with Astral loss is usually tight enough to be informative, e.g., for a highly anisotropic equation, on average, Astral overestimates error by a factor of $1.5$, and for convection-diffusion by a factor of $1.7$.