On Loss Functionals for Physics-Informed Neural Networks for Steady-State Convection-Dominated Convection-Diffusion Problems

Abstract

Solutions of convection-dominated convection-diffusion problems usually possess layers, which are regions where the solution has a steep gradient. It is well known that many classical numerical discretization techniques face difficulties when approximating the solution to these problems. In recent years, physics-informed neural networks (PINNs) for approximating the solution to (initial-)boundary value problems ((I)BVPs) received a lot of interest. This paper studies various loss functionals for PINNs that are especially designed for convection-dominated convection-diffusion problems and that are novel in the context of PINNs. They are numerically compared to the vanilla and an hp-variational loss functional from the literature based on two steady-state benchmark problems whose solutions possess different types of layers. We observe that the best novel loss functionals reduce the $L^2(\varOmega )$ error by 17.3% for the first and 5.5% for the second problem compared to the methods from the literature.

Funding

Weierstraß-Institut für Angewandte Analysis und Stochastik, Leibniz-Institut im Forschungsverbund Berlin e.V. (3510)