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The model reduction of a mesoscopic kinetic dynamics to a macroscopiccontinuum dynamics has been one of the fundamental questions in mathematicalphysics since Hilbert's time. In this paper, we consider a diagram of thediffusion limit from the Vlasov-Poisson-Fokker-Planck (VPFP) system on abounded interval with the specular reflection boundary condition to thePoisson-Nernst-Planck (PNP) system with the no-flux boundary condition. Weprovide a Deep Learning algorithm to simulate the VPFP system and the PNPsystem by computing the time-asymptotic behaviors of the solution and thephysical quantities. We analyze the convergence of the neural network solutionof the VPFP system to that of the PNP system via the Asymptotic-Preserving (AP)scheme. Also, we provide several theoretical evidence that the Deep NeuralNetwork (DNN) solutions to the VPFP and the PNP systems converge to the apriori classical solutions of each system if the total loss function vanishes.

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