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Maxwell-Amp\`{e}re-Nernst-Planck (MANP) equations were recently proposed tomodel the dynamics of charged particles. In this study, we enhance a numericalalgorithm of this system with deep learning tools. The proposed hybridalgorithm provides an automated means to determine a proper approximation forthe dummy variables, which can otherwise only be obtained through massivenumerical tests. In addition, the original method is validated for2-dimensional problems. However, when the spatial dimension is one, theoriginal curl-free relaxation component is inapplicable, and the approximationformula for dummy variables, which works well in a 2-dimensional scenario,fails to provide a reasonable output in the 1-dimensional case. The proposedmethod can be readily generalised to cases with one spatial dimension.Experiments show numerical stability and good convergence to the steady-statesolution obtained from Poisson-Boltzmann type equations in the 1-dimensionalcase. The experiments conducted in the 2-dimensional case indicate that theproposed method preserves the conservation properties.

A conservative hybrid physics-informed neural network method for Maxwell-Amp\`{e}re-Nernst-Planck equations