Stabilized finite element methods for convection-dominated transport problems -- where sharp fronts, boundary layers, and shocks appear -- introduce numerical diffusion to prevent oscillations. This stabilization is necessary but inexact: it smears features that should remain sharp. Cengizci, Ugur, and Natesan propose a hybrid framework that applies physics-informed neural network corrections selectively to the final temporal snapshots of a stabilized finite element simulation, rather than training a neural network across the entire space-time domain.
The stabilization backbone uses Streamline-Upwind Petrov-Galerkin (SUPG) augmented with a YZbeta shock-capturing operator to handle the bulk of the computation. The neural network then takes the last K_s temporal snapshots as input and applies a correction constrained by residuals of the governing equations and boundary conditions. The architecture employs residual blocks with random Fourier features and progressive training with adaptive loss weighting. Across five benchmark problems -- including boundary layers, interior layers, traveling waves, and nonlinear Burgers dynamics -- the corrected solutions show significant accuracy improvements at the terminal time compared to the finite element solution alone.
The design principle is temporal selectivity. Rather than asking a neural network to learn the full dynamics -- which is expensive and often poorly conditioned for convection-dominated flows -- the method confines the correction to where accumulated numerical diffusion is largest: the end of the simulation. The finite element method handles the well-posed parts of the problem, and the neural network handles the residual error where it has concentrated. This division of labor exploits the fact that stabilization errors accumulate systematically, making them predictable targets for a learned correction.
(arXiv:2603.03259)