PNP 方程的基于高斯过程回归的新 Gummel 迭代算法

数学物理学报 ›› 2024, Vol. 44 ›› Issue (5): 1302-1310.

PNP 方程的基于高斯过程回归的新 Gummel 迭代算法

敖渝焱,阳莺^*()

桂林电子科技大学数学与计算科学学院 & 广西应用数学中心 (GUET) & 广西高校数据分析与计算重点实验室广西桂林 541004

收稿日期:2023-12-13 修回日期:2024-04-15 出版日期:2024-10-26 发布日期:2024-10-16
通讯作者: *阳莺, E-mail: yangying@lsec.cc.ac.cn
基金资助:
国家自然科学基金(12161026);广西科技基地和人才专项(AD23026048);广西自然科学基金(2020GXNSFAA159098);广西科技项目(AD23023002)

A New Gummel Iterative Algorithm Based on Gaussian Process Regression for the PNP Equation

Ao Yuyan,Yang Ying^*()

Guilin University of Electronic Technology, School of Mathematics and Computating Science & Guangxi Applied Mathematics Center (GUET) & Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, Guangxi Guilin 541004

Received:2023-12-13 Revised:2024-04-15 Online:2024-10-26 Published:2024-10-16
Supported by:
NSFC(12161026);Special Fund for Scientific and Technological Bases and Talents of Guangxi(AD23026048);Guangxi Natural Science Foundation(2020GXNSFAA159098);Science and Technology Project of Guangxi(AD23023002)

摘要/Abstract

摘要：

Poisson-Nernst-Planck (PNP) 方程是由 Poisson 方程和 Nernst-Planck 方程耦合而成的一类非线性偏微分方程组, 其常用的线性化迭代方法-Gummel 迭代的效率很大程度上受松弛参数的影响. 机器学习中的高斯过程回归 (GPR) 方法因其训练规模较小, 且不需要提供函数关系, 在该文中被应用于预测 Gummel 迭代的较优松弛参数, 加速迭代的收敛速度. 首先针对 PNP 方程的 Gummel 迭代, 设计了一种可预测松弛参数的 GPR 方法. 其次利用 Box-Cox 转换方法, 对 Gummel 迭代的数据进行预处理, 提高 GPR 方法的准确性. 最后基于 GPR 方法及 Box-Cox 转换算法, 提出了 PNP 方程的一种新的 Gummel 迭代算法. 数值实验表明, 新 Gummel 迭代算法与经典的 Gummel 迭代算法相比, 求解效率更高, 且收敛阶相同.

关键词: Poisson-Nernst-Planck 方程, Gummel 迭代, 高斯过程回归, 参数预测, 机器学习

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

中图分类号:

O241.82

敖渝焱, 阳莺. PNP 方程的基于高斯过程回归的新 Gummel 迭代算法[J]. 数学物理学报, 2024, 44(5): 1302-1310.

Ao Yuyan, Yang Ying. A New Gummel Iterative Algorithm Based on Gaussian Process Regression for the PNP Equation[J]. Acta mathematica scientia,Series A, 2024, 44(5): 1302-1310.

图/表 4

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