多元多项式函数的三层前向神经网络逼近方法

该文首先用构造性方法证明:对任意r阶多元多项式,存在确定权值和确定隐元个数的三层前向神经网络.它能以任意精度逼近该多项式.其中权值由所给多元多项式的系数和激活函数确定,而隐元个数由r与输入变量维数确定.作者给出算法和算例,说明基于文中所构造的神经网络可非常高效地逼近多元多项式函数.具体化到一元多项式的情形,文中结果比曹飞龙等所提出的网络和算法更为简单、高效;所获结果对前向神经网络逼近多元多项式函数类的网络构造以及逼近等具有重要的理论与应用意义,为神经网络逼近任意函数的网络构造的理论与方法提供了一条途径.

Approximation Method of Multivariate Polynomials by Feedforward Neural Networks

Firstly,this paper investigates that for a given multivariate polynomials with r order,a three-layer feedforward neural networks with determinate weights and the number of hiddenlayer nodes can be established by a constructive method to approximate the polynomials to any degree of accuracy.Secondly,the weights are decided by both the coefficients of the polynomials and the activation function,and the number of hidden-layer nodes of the constructed network depends on the order of approximating polynomial and the dimension of input on the network.Then the algorithm and algorithmic examples are given,where the constructed networks can very efficiently approximate multivariate polynomials.Specifically,for a univariate polynomial,the constructed network and realization of algorithm obtained are simpler and more efficient than the methods proposed by Cao Fei-Long in 2003.The obtained results are of theoretical and practical importance in constructing a feedforward neural network with three-layer to approximate the class of multivariate polynomials.They also provide a route in both theory and method of constructing neural network to approximate any multivariate functions.