基于前馈型神经网络解变系数分数阶积分微分方程  

Solving fractional integro-differential equation with variable coefficients based on feedforward neural network

作　　者：杨刘盼 郭安祺 邵新平 YANG Liupan;GUO Anqi;SHAO Xinping(School of Sciences,Hangzhou Dianzi University,Hangzhou Zhejiang 310018,China)

机构地区：[1]杭州电子科技大学理学院,浙江杭州310018

出　　处：《杭州电子科技大学学报（自然科学版）》2023年第3期76-82,共7页Journal of Hangzhou Dianzi University：Natural Sciences

基　　金：国家自然科学基金资助项目(11701133)。

摘　　要：为了研究变系数分数阶积分微分方程的数值解,提出了一种基于Bernstein多项式的前馈型神经网络求解变系数分数阶Fredholm积分微分方程的方法。首先,根据Caputo分数阶导数的定义,将变系数分数阶的积分微分方程转化为Bernstein多项式空间上的矩阵形式;然后,将Bernstein多项式的系数作为权重,构造前馈型神经网络,采用梯度下降法对权重进行学习,从而得到近似解;接着,从理论上证明了该前馈型神经网络的收敛性;最后,通过数值实例分析验证了提出方法的有效性。In order to study the numerical solutions of fractional integro-differential equation with variable coefficients,a feedforward neural network based on Bernstein polynomials is proposed to solve fractional Fredholm integro-differential equation with variable coefficients.According to the definition of Caputo's fractional derivative,the Fractional integro-differential equation with variable coefficients is converted into the matrix form in Bernstein's polynomial space.The feedforward neural network is constructed by taking the coefficients of Bernstein polynomials as weights of the network.The approximate solution is obtained by gradient descent method and the convergence of the neural network is proved theoretically.Finally,numerical examples are given to illustrate the effectiveness of the proposed method.

关 键 词：BERNSTEIN多项式 神经网络 分数阶积分微分 变系数 

分 类 号：O24[理学—计算数学]