Work

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

Year: 2024

Type: article

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 $$L^2(\varOmega )$$ L 2 ( Ω ) error by 17.3% for the first and 5.5% for the second problem compared to the methods from the literature.

Source: Communications on Applied Mathematics and Computation

DOI: 10.1007/s42967-024-00433-7

Authors: Derk Frerichs-Mihov, Linus Henning, Volker John

Institutions: Weierstrass Institute for Applied Analysis and Stochastics, Freie Universität Berlin

Language: English

Topic: Model Reduction and Neural Networks

Subfield: Statistical and Nonlinear Physics

Field: Physics and Astronomy

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,090

Funder: Weierstraß-Institut