论文标题
通过深神经网络方法，将Vlasov-Poisson-Fokker-Planck系统降低到Poisson-Nernst-Planck系统
The model reduction of the Vlasov-Poisson-Fokker-Planck system to the Poisson-Nernst-Planck system via the Deep Neural Network Approach
论文作者
论文摘要
自希尔伯特（Hilbert）时代以来，介质动力学的模型还原为宏观连续性动力学一直是数学物理学的基本问题之一。在本文中，我们考虑了从vlasov-poisson-fokker-planck（VPFP）系统的扩散极限的图表，该系统在具有无液光边界条件的Poisson-Nernst-Planck（PNP）系统的侧面反射边界条件上均具有侧面反射边界条件。我们提供了一种深度学习算法，以通过计算解决方案和物理量的时间 - 反应行为来模拟VPFP系统和PNP系统。我们通过渐近保护（AP）方案分析了VPFP系统的神经网络解决方案与PNP系统的收敛性。此外，我们提供了一些理论上的证据，即VPFP的深神经网络（DNN）解决方案，而PNP系统将如果总损耗函数消失，则将PNP系统收敛到每个系统的先验经典解决方案。
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.
