The model reduction of the Vlasov–Poisson–Fokker–Planck system to the Poisson–Nernst–Planck system via the Deep Neural Network Approach

Jae Yong Lee,Jin Woo Jang,Hyung Ju Hwang

doi:10.1051/m2an/2021038

Abstract

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.

Highlights

The aim of this paper is to establish the commutation of the following diagram of diffusion limit, which provides the reduction of the kinetic equation to the fluid equation as the perturbation parameter ε tends to zero: Keywords and phrases
We have introduced the Deep Neural Network (DNN) solutions to the VPFP system and the PNP system using the Deep Learning algorithm
We propose appropriate loss functions for training, including the loss function for the initial conditions and the boundary conditions to each system: the VPFP system in Part II and the PNP system in Part III

Summary

Introduction

The description of the evolution of gases has been explained via the statistical approach on the probabilistic distribution functions on the mesoscopic level, whereas the fluid theory describes the dynamics on the macroscopic level. Each of these interpretations and the asymptotic expansions of the mesoscopic equations to the macroscopic equations have been crucial issues. The aim of this paper is to establish the commutation of the following diagram of diffusion limit, which provides the reduction of the kinetic equation (the Vlasov–Poisson–Fokker–Planck system) to the fluid equation (the Poisson–Nernst–Planck system) as the perturbation parameter ε tends to zero:.

Objectives

Methods

Conclusion