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.
Accepté le :
Publié le :
DOI : 10.1051/m2an/2021038
Keywords: Vlasov–Poisson–Fokker–Planck system, Poisson–Nernst–Planck system, diffusion limit, artificial neural network, asymptotic-preserving scheme
@article{M2AN_2021__55_5_1803_0,
author = {Lee, Jae Yong and Jang, Jin Woo and Hwang, Hyung Ju},
title = {The model reduction of the {Vlasov{\textendash}Poisson{\textendash}Fokker{\textendash}Planck} system to the {Poisson{\textendash}Nernst{\textendash}Planck} system \protect\emph{via} the {Deep} {Neural} {Network} {Approach}},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1803--1846},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
number = {5},
doi = {10.1051/m2an/2021038},
mrnumber = {4313375},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2021038/}
}
TY - JOUR AU - Lee, Jae Yong AU - Jang, Jin Woo AU - Hwang, Hyung Ju TI - The model reduction of the Vlasov–Poisson–Fokker–Planck system to the Poisson–Nernst–Planck system via the Deep Neural Network Approach JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2021 SP - 1803 EP - 1846 VL - 55 IS - 5 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2021038/ DO - 10.1051/m2an/2021038 LA - en ID - M2AN_2021__55_5_1803_0 ER -
%0 Journal Article %A Lee, Jae Yong %A Jang, Jin Woo %A Hwang, Hyung Ju %T The model reduction of the Vlasov–Poisson–Fokker–Planck system to the Poisson–Nernst–Planck system via the Deep Neural Network Approach %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2021 %P 1803-1846 %V 55 %N 5 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2021038/ %R 10.1051/m2an/2021038 %G en %F M2AN_2021__55_5_1803_0
Lee, Jae Yong; Jang, Jin Woo; Hwang, Hyung Ju. The model reduction of the Vlasov–Poisson–Fokker–Planck system to the Poisson–Nernst–Planck system via the Deep Neural Network Approach. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 5, pp. 1803-1846. doi: 10.1051/m2an/2021038
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