On loss functionals for physics-informed neural networks for convection-dominated convection-diffusion problems

Abstract

In the convection-dominated regime, solutions of convection-diffusion problems usually possesses 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 received a lot of interest. In this work, we study various loss functionals for PINNs that are novel in the context of PINNs and are especially designed for convection-dominated convection-diffusion problems. They are numerically compared to the vanilla and a $hp$-variational loss functional from the literature based on two benchmark problems whose solutions possess different types of layers. We observe that the best novel loss functionals reduce the $L^2(Omega)$ error by $17.3%$ for the first and $5.5%$ for the second problem compared to the methods from the literature.