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

doi:10.1007/s42967-024-00433-7

Communications on Applied Mathematics and Computation ›› 2026, Vol. 8 ›› Issue (1): 287-308.doi: 10.1007/s42967-024-00433-7

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

Derk Frerichs-Mihov¹, Linus Henning², Volker John^1,2

1. Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Mohrenstr. 39, 10117, Berlin, Germany;
2. Department of Mathematics and Computer Science, Free University Berlin, Arnimallee 6, 14195, Berlin, Germany

收稿日期:2023-12-08 修回日期:2024-04-23 出版日期:2026-02-20 发布日期:2026-02-11
通讯作者: Volker John,E-mail:john@wias-berlin.de E-mail:john@wias-berlin.de
作者简介:Derk Frerichs-Mihov,E-mail:frerichs-mihov@wias-berlin.de;Linus Henning,E-mail:linus.henning@fu-berlin.de
基金资助:
Open Access funding enabled and organized by Projekt DEAL.

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

Derk Frerichs-Mihov¹, Linus Henning², Volker John^1,2

1. Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Mohrenstr. 39, 10117, Berlin, Germany;
2. Department of Mathematics and Computer Science, Free University Berlin, Arnimallee 6, 14195, Berlin, Germany

Received:2023-12-08 Revised:2024-04-23 Online:2026-02-20 Published:2026-02-11
Contact: Volker John,E-mail:john@wias-berlin.de E-mail:john@wias-berlin.de
Supported by:
Open Access funding enabled and organized by Projekt DEAL.

摘要/Abstract

摘要： 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 L²(Ω) error by 17.3% for the first and 5.5% for the second problem compared to the methods from the literature.

关键词: Steady-state convection-diffusion problems, Convection-dominated regime, Physics-informed neural networks (PINNs), Loss functionals

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 L²(Ω) error by 17.3% for the first and 5.5% for the second problem compared to the methods from the literature.

Key words: Steady-state convection-diffusion problems, Convection-dominated regime, Physics-informed neural networks (PINNs), Loss functionals

中图分类号:

Derk Frerichs-Mihov, Linus Henning, Volker John. On Loss Functionals for Physics-Informed Neural Networks for Steady-State Convection-Dominated Convection-Diffusion Problems[J]. Communications on Applied Mathematics and Computation, 2026, 8(1): 287-308.

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

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

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