PNP方程的基于高斯过程回归的新Gummel迭代算法  

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

作　　者：敖渝焱 阳莺[1] Ao Yuyan;Yang Ying(Guilin University of Electronic Technology,School of Mathematics and Computating Science&Guangri Applied Mathematics Center(GUET)&Guangri Colleges and Universities Key Laboratory of Data Analysis and Computation,Guangci Guilin 541004)

机构地区：[1]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

出　　处：《数学物理学报（A辑）》2024年第5期1302-1310,共9页Acta Mathematica Scientia

基　　金：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)。

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.PNP Equations are a class of nonlinear partial differential equations coupled from Poisson and Nernst planck equations,and the eficiency 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.

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 

分 类 号：O241.82[理学—计算数学]