A conservative hybrid physics-informed neural network method for Maxwell-Ampère-Nernst-Planck equations
This work addresses a domain-specific problem in computational physics for researchers modeling charged particles, offering an incremental improvement by generalizing an existing method to 1D cases.
The study tackled the challenge of modeling charged particle dynamics with Maxwell-Ampère-Nernst-Planck equations by enhancing a numerical algorithm with deep learning tools, achieving numerical stability and good convergence to steady-state solutions in 1D and preserving conservation properties in 2D.
Maxwell-Ampère-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.