On loss functionals for physics-informed neural networks for convection-dominated convection-diffusion problems
Authors
- Frerichs-Mihov, Derk
ORCID: 0000-0002-4474-5042 - Henning, Linus
- John, Volker
ORCID: 0000-0002-2711-4409
2020 Mathematics Subject Classification
- 65N99 68T07
Keywords
- Convection-diffusion problems, convection-dominated regime, physics-informed neural networks, loss functionals
DOI
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.
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