A conservative hybrid physics-informed neural network method for
  Maxwell-Ampère-Nernst-Planck equations
Cheng Chang Zhouping Xin Tieyong Zeng
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.