Trend to equilibrium for the kinetic Fokker-Planck equation via the neural network approach

JOURNAL OF COMPUTATIONAL PHYSICS

卷:419

Article

Hwang, Hyung Ju¹  Jang, Jin Woo²  Jo, Hyeontae¹  Lee, Jae Yong¹ 

[1] Pohang Univ Sci & Technol POSTECH, Dept Math, Pohang 37673, South Korea

[2] Inst Basic Sci IBS, Ctr Geometry & Phys, Pohang 37673, South Korea

关键词: Fokker-Planck equation;    Asymptotic behavior of solutions;    Kinetic theory of gases;    Artificial intelligence;   

DOI  :  10.1016/j.jcp.2020.109665

来源: Elsevier

【 摘 要 】

The issue of the relaxation to equilibrium has been at the core of the kinetic theory of rarefied gas dynamics. In the paper, we introduce the Deep Neural Network (DNN) approximated solutions to the kinetic Fokker-Planck equation in a bounded interval and study the large-time asymptotic behavior of the solutions and other physically relevant macroscopic quantities. We impose the varied types of boundary conditions including the inflow-type and the reflection-type boundaries as well as the varied diffusion and friction coefficients and study the boundary effects on the asymptotic behaviors. These include the predictions on the large-time behaviors of the pointwise values of the particle distribution and the macroscopic physical quantities including the total kinetic energy, the entropy, and the free energy. We also provide the theoretical supports for the pointwise convergence of the neural network solutions to the a priorianalytic solutions. We use the library PyTorch, the activation function tanhbetween layers, and the Adamoptimizer for the Deep Learning algorithm. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free