The model reduction of the Vlasov-Poisson-Fokker-Planck system to the Poisson-Nernst-Planck system via the Deep Neural Network Approach
This addresses a fundamental question in mathematical physics for researchers in kinetic theory and computational modeling, but it appears incremental as it applies existing deep learning methods to a specific reduction problem.
The paper tackles the model reduction problem from the Vlasov-Poisson-Fokker-Planck (VPFP) system to the Poisson-Nernst-Planck (PNP) system using a deep learning algorithm, analyzing convergence via an Asymptotic-Preserving scheme and providing theoretical evidence for solution accuracy.
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.