Astral: training physics-informed neural networks with error majorants

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.

💡 Research Summary

The paper addresses a fundamental limitation of current physics‑informed neural networks (PiNNs): the standard practice of minimizing the PDE residual provides only an indirect and often unreliable measure of the true solution error. Through simple analytical examples and a systematic statistical study on diffusion problems, the authors demonstrate that residual magnitude correlates poorly with the energy‑norm error (correlation ≈ 0.2), and can be arbitrarily large even when the actual error is negligible, or zero while the error is substantial.

To overcome this, the authors propose a new loss function called Astral (NeurAl a poSTerioR i functionAl Loss), which is built directly from a functional a‑posteriori error estimate (error majorant). The key idea is to augment the neural network so that it predicts not only an approximate solution (e_\phi) but also an auxiliary flux field (e_F) (or other problem‑specific auxiliary variables). By introducing a free auxiliary field (w) and problem‑dependent constants (\alpha) and (\beta), the majorant takes the generic form

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