Volume 55, Number 5, September-October 2021
|1803 - 1846
|17 September 2021
The model reduction of the Vlasov–Poisson–Fokker–Planck system to the Poisson–Nernst–Planck system via the Deep Neural Network Approach
Department of Mathematics, Pohang University of Science and Technology (POSTECH), Pohang 37673, Republic of Korea
2 Institute for Applied Mathematics, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany
* Corresponding author: firstname.lastname@example.org
Accepted: 27 July 2021
The model reduction of a mesoscopic kinetic dynamics to a macroscopic continuum dynamics has been one of the fundamental questions in mathematical physics since Hilbert’s time. In this paper, we consider a diagram of the diffusion limit from the Vlasov–Poisson–Fokker–Planck (VPFP) system on a bounded interval with the specular reflection boundary condition to the Poisson–Nernst–Planck (PNP) system with the no-flux boundary condition. We provide a Deep Learning algorithm to simulate the VPFP system and the PNP system by computing the time-asymptotic behaviors of the solution and the physical quantities. We analyze the convergence of the neural network solution of the VPFP system to that of the PNP system via the Asymptotic-Preserving (AP) scheme. Also, we provide several theoretical evidence that the Deep Neural Network (DNN) solutions to the VPFP and the PNP systems converge to the a priori classical solutions of each system if the total loss function vanishes.
Mathematics Subject Classification: 68T20 / 35Q84 / 35B40 / 82C40
Key words: Vlasov–Poisson–Fokker–Planck system / Poisson–Nernst–Planck system / diffusion limit / artificial neural network / asymptotic-preserving scheme
© The authors. Published by EDP Sciences, SMAI 2021
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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