Abstract:

PNP Equations are a class of nonlinear partial differential equations coupled from Poisson and Nernst planck equations, and the efficiency of its Gummel iteration, a commonly used linearization iteration method, is largely affected by the relaxation parameter. The Gaussian Process Regression (GPR) method in machine learning, due to its small training size and the fact that it does not need to provide a functional relationship, is applied in that paper to predict the preferred relaxation parameters for the Gummel iteration and accelerate the convergence of the iteration. Firstly GPR method with predictable relaxation parameters is designed for the Gummel iteration of the PNP equation. Secondly, the Box-Cox transformation method is utilized to preprocess the data of Gummel iteration to improve the accuracy of the GPR method. Finally, based on the GPR method and Box-Cox transformation algorithm, a new Gummel iteration algorithm for the PNP equation is proposed. Numerical experiments show that the new Gummel iterative algorithm is more efficient in solving and has the same convergence order compared to the classical Gummel iterative algorithm.

Key words: Poisson-Nernst-Planck equations, Gummel iteration, Gaussian process regression, Parameter prediction, Machine learning