A conservative hybrid physics-informed neural network method for Maxwell-Amp\`{e}re-Nernst-Planck equations
Abstract
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.
Keywords
Cite
@article{arxiv.2312.05891,
title = {A conservative hybrid physics-informed neural network method for Maxwell-Amp\`{e}re-Nernst-Planck equations},
author = {Cheng Chang and Zhouping Xin and Tieyong Zeng},
journal= {arXiv preprint arXiv:2312.05891},
year = {2023}
}