Astral: training physics-informed neural networks with error majorants

TL;DR

This paper introduces Astral, a novel loss function for physics-informed neural networks that directly estimates error bounds, enabling more reliable training and error assessment compared to traditional residual minimization methods.

Contribution

The paper proposes a new error majorant-based loss function, Astral, for physics-informed neural networks, improving error estimation and convergence speed over residual-based approaches.

Findings

Astral loss provides reliable error estimates with tight bounds.

Astral loss leads to faster convergence and lower errors.

Error estimates from Astral are better correlated with actual error.

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{R}i function\textbf{A}l \textbf{L}oss. 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, magnetostatics and nonlinear elastoplasticity problems. The results indicate that Astral loss is competitive to the residual loss, typically leading to faster convergence and lower error. The main benefit of using Astral loss comes from its ability to estimate error, which is impossible with other loss functions. Our experiments indicate that the error estimate obtained with Astral loss is usually tight enough, 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 $1.7$. We further demonstrate that Astral loss is better correlated with error than residual and is a more reliable predictor of the error value. Moreover, unlike residual, the error indicator obtained from Astral loss has a superb spatial correlation with error. Backed with the empirical and theoretical results, we argue that one can productively use Astral loss to perform reliable error analysis and approximate PDE solutions with accuracy similar to standard residual-based techniques.