基于正交Legendre多项式的神经网络求解随机Itô-Volterra积分方程  

Neural Network Based on Orthogonal Legendre Polynomials for Solving Stochastic Itô-Volterra Integral Equation

作　　者：曾嵘 张洪铭 邵新平 ZENG Rong;ZHANG Hongming;SHAO Xin-ping(School of Sciance Hangzhou Dianzi University,Hangzhou 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）》2024年第4期80-87,共8页Journal of Hangzhou Dianzi University：Natural Sciences

基　　金：国家自然科学基金项目(11701133)。

摘　　要：为求多维随机Itô-Volterra积分方程近似解,首先利用Legendre多项式的函数逼近和带配点法的算子矩阵,将所求方程转化为代数方程组。基于算子矩阵构造损失函数,并用配置点作为网络输入、Legendre多项式的系数作为权重构造神经网络,再采用梯度下降法对权重进行学习从而得到近似解。从理论上对该方法的收敛性进行了分析。最后,通过数值算例验证该方法的准确性。In order to solve the approximate solution of the multi-dimensional stochastic Itô-Volterra integral equation,the Legendre polynomial function approximation and the operation matrix with ligand method are used to convert the stochastic Itô-Volterra integral equation into algebraic equations.The loss function is constructed based on the operator matrix,and the neural network is constructed by using the configuration points as the network input and the coefficients of the Legendre polynomials as the weights.The approximate solutions are obtained by using the gradient descent method to learn the weights,and the corresponding convergence is analyzed theoretically.Finally,numerical examples are given to verify the accuracy of the proposed method.

关 键 词：LEGENDRE多项式 算子矩阵 神经网络 随机积分方程 

分 类 号：O24[理学—计算数学]