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

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Title
Trend to equilibrium for the kinetic Fokker-Planck equation via the neural network approach
Author(s)
Hwang H.J.; Jang Jin Woo; Jo H.; Lee J.Y.
Subject
Artificial intelligence, ; Asymptotic behavior of solutions, ; Fokker-Planck equation, ; Kinetic theory of gases
Publication Date
2020-10-15
Journal
JOURNAL OF COMPUTATIONAL PHYSICS, v.419, pp.109665
Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
Abstract
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.
URI
https://pr.ibs.re.kr/handle/8788114/7621
DOI
10.1016/j.jcp.2020.109665
ISSN
0021-9991
Appears in Collections:
Center for Geometry and Physics(기하학 수리물리 연구단) > 1. Journal Papers (저널논문)
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