Stabilizing convection-dominated flow problems using neural networks based on flux-limiting techniques

Abstract

Convection-dominated flow problems are well-known to have non-physical oscillations near steep gradients or discontinuities in the solution when solved with standard numerical methods, such as finite elements or finite difference methods. To overcome this limitation, algebraic flux correction (AFC) can be used, which is a stabilization method. However, AFC contains time-consuming computations, therefore, alternative approaches are explored. The rapidly rising field of machine learning in the mathematical world, so called scientific machine learning, has successful applications in solving partial differential equations. In this work, the focus is on convection-dominated flow problems, in particular the steady state convection-diffusion equation in one-dimension. To solve this, two alternative approaches based on neural network-learning have been developed that are able to mimic the AFC limiter with a certain accuracy and performance. In some cases, the neural network-based limiter is outperforming the AFC limiter.