Trend to equilibrium for the kinetic Fokker-Planck equation via the neural network approach
© 2020 Elsevier Inc. All rights reserved.
| Veröffentlicht in: | Journal of computational physics. - 1986. - 419(2020) vom: 15. Okt., Seite 109665 |
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| Weitere Verfasser: | , , |
| Format: | Online-Aufsatz |
| Sprache: | English |
| Veröffentlicht: |
2020
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| Zugriff auf das übergeordnete Werk: | Journal of computational physics |
| Schlagworte: | Journal Article Artificial intelligence Asymptotic behavior of solutions Fokker-Planck equation Kinetic theory of gases |
| Zusammenfassung: | © 2020 Elsevier Inc. All rights reserved. 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 priori analytic solutions. We use the library PyTorch, the activation function tanh between layers, and the Adam optimizer for the Deep Learning algorithm |
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| Beschreibung: | Date Revised 28.03.2024 published: Print-Electronic Citation Status PubMed-not-MEDLINE |
| ISSN: | 0021-9991 |
| DOI: | 10.1016/j.jcp.2020.109665 |