Maxwell-Amp\`{e}re-Nernst-Planck (MANP) equations were recently proposed to model the dynamics of charged particles. In this study, we enhance a numerical algorithm of this system with deep learning tools. The proposed hybrid algorithm provides an automated means to determine a proper approximation for the dummy variables, which can otherwise only be obtained through massive numerical tests. In addition, the original method is validated for 2-dimensional problems. However, when the spatial dimension is one, the original curl-free relaxation component is inapplicable, and the approximation formula for dummy variables, which works well in a 2-dimensional scenario, fails to provide a reasonable output in the 1-dimensional case. The proposed method can be readily generalised to cases with one spatial dimension. Experiments show numerical stability and good convergence to the steady-state solution obtained from Poisson-Boltzmann type equations in the 1-dimensional case. The experiments conducted in the 2-dimensional case indicate that the proposed method preserves the conservation properties.

翻译：Maxwell-Ampère-Nernst-Planck（MANP）方程近期被提出用于模拟带电粒子的动力学行为。本研究利用深度学习工具对该系统的数值算法进行了改进。所提出的混合算法提供了一种自动确定虚拟变量适当近似值的方法，而此前这类近似只能通过大量数值试验获得。此外，原始方法仅适用于二维问题的验证。然而当空间维度为一维时，原始的无旋松弛分量无法应用，且虚拟变量的近似公式（在二维场景中表现良好）在一维情形下无法输出合理结果。本方法可便捷地推广至一维空间情况。实验表明，在一维情形下，该方法具有数值稳定性，并能良好收敛至由Poisson-Boltzmann型方程获得的稳态解。二维情形的实验证明，所提方法保持了守恒性质。