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Derk Frerichs-Mihov, Volker John, Marwa Zainelabdeen
On collocation points for physics-informed neural networks applied to convection-dominated convection-diffusion problems, Proceedings ENUMATH 2023, accepted, 2024
In recent years physics-informed neural networks (PINNs) for approximating the solution to (initial-)boundary value problems gained a lot of interest. PINNs are trained to minimize several residuals of the problem in collocation points. In this work we tackle convection-dominated convection-diffusion problems, whose solutions usually possess layers, which are small regions where the solution has a steep gradient. Inspired by classical Shishkin meshes, we compare hard-constrained PINNs trained with layer-adapted collocation points with ones trained with equispaced and uniformly randomly chosen points. We observe that layer-adapted points work the best for a problem with an interior layer and the worst for a problem with boundary layers. For both problems at most acceptable solutions can be obtained with PINNs.
Derk Frerichs-Mihov, Linus Henning, Volker John
On loss functionals for physics-informed neural networks for steady-state convection-dominated convection-diffusion problems, Communications on Applied Mathematics and Computation, accepted, 2024
| Solutions of convection-dominated 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. 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 a $hp$-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 $L^2(\Omega)$ error by $17.3\%$ for the first and $5.5\%$ for the second problem compared to the methods from the literature.
Derk Frerichs-Mihov, Linus Henning, Volker John
Using deep neural networks for detecting spurious oscillations in discontinuous Galerkin solutions of convection-dominated convection-diffusion equations, J. Sci. Comp. 97, Article 36, 2023
| Standard discontinuous Galerkin (DG) finite element solutions to convection-dominated con\-vec\-tion-diffusion equations usually possess sharp layers but also exhibit large spurious oscillations. Slope limiters are known as a post-processing technique to reduce these unphysical values. This paper studies the application of deep neural networks for detecting mesh cells on which slope limiters should be applied. The networks are trained with data obtained from simulations of a standard benchmark problem with linear finite elements. It is investigated how they perform when applied to discrete solutions obtained with higher order finite elements and to solutions for a different benchmark problem. |