Purpose

This paper aims to develop a novel algorithm and apply it to solve two-dimensional linear partial differential equations (PDEs). The proposed method is based on Chebyshev neural network and extreme learning machine (ELM) called Chebyshev extreme learning machine (Ch-ELM) method.

Design/methodology/approach

The network used in the proposed method is a single hidden layer feedforward neural network. The Kronecker product of two Chebyshev polynomials is used as basis function. The weights from the input layer to the hidden layer are fixed value 1. The weights from the hidden layer to the output layer can be obtained by using ELM algorithm to solve the linear equations established by PDEs and its definite conditions.

Findings

To verify the effectiveness of the proposed method, two-dimensional linear PDEs are selected and its numerical solutions are obtained by using the proposed method. The effectiveness of the proposed method is illustrated by comparing with the analytical solutions, and its superiority is illustrated by comparing with other existing algorithms.

Originality/value

Ch-ELM algorithm for solving two-dimensional linear PDEs is proposed. The algorithm has fast execution speed and high numerical accuracy.

中文翻译：

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极限学习机算法求解基于Chebyshev神经网络的二维线性偏微分方程

目的

本文旨在开发一种新颖的算法并将其应用于求解二维线性偏微分方程（PDE）。所提出的方法基于Chebyshev神经网络和极限学习机（ELM），称为Chebyshev极限学习机（Ch-ELM）方法。

设计/方法/方法

所提出的方法中使用的网络是单隐藏层前馈神经网络。两个Chebyshev多项式的Kronecker积用作基函数。从输入层到隐藏层的权重是固定值1。从隐藏层到输出层的权重可以通过使用ELM算法求解由PDE建立的线性方程及其确定条件来获得。

发现

为了验证该方法的有效性，选择了二维线性PDE，并通过该方法获得了数值解。通过与解析解的比较说明了所提方法的有效性，并与其他现有算法进行了比较说明了其优越性。

创意/价值

提出了求解二维线性PDE的Ch-ELM算法。该算法执行速度快，数值精度高。