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A conservative hybrid physics-informed neural network method for Maxwell-Amp`ere-Nernst-Planck equations

arXiv 2312.05891 ↗First posted 10 Dec 2023math.NA cs.LG cs.NAcited by 0

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Abstract (arXiv)

Maxwell-Amp`ere-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.

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References · 14

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  11. 2020DiscretizationNet: A Machine-Learning based solver for Navier-Stokes Equations using Finite Volume Discretizationcited by 35
  12. 2021Hybrid FEM-NN models: Combining artificial neural networks with the finite element methodcited by 24
2 more
  1. 2020An unsupervised learning approach to solving heat equations on chip based on Auto Encoder and Image Gradientcited by 13
  2. 2020The model reduction of the Vlasov-Poisson-Fokker-Planck system to the Poisson-Nernst-Planck system via the Deep Neural Network Approachcited by 6

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