多项式函数的神经网络逼近: 网络的构造与逼近算法

该文作者先用构造性方法证明：对于给定的r阶多项式函数，可以具体地构造出一个三层前向神经网络，以任意精度逼近该多项式，所构造的网络的隐层节点个数仅与多项式的阶数r和网络的输入个数s有关，并能准确地用r表达；然后，给出一个实现这一逼近的具体算法；最后，给出两个数值算例进一步验证所得的理论结果．该文所获得的结果对前向神经网络逼近多项式函数类的网络具体构造以及实现逼近的方法等问题具有较为重要的指导意义．

Approximation of Polynomial Functions by Neural Network:Construction of Network and Algorithm of Approximation

It is investigated in this paper that the polynomial functions are approximated by feed-forward neural network with three-layer. Firstly, It is shown that for a given polynomial function with r order a feed-forward neural network with three-layer can be constructed by a constructive method to approximate the polynomial to any degree of accuracy. The number of hidden-layer nodes of the constructed network only depends on the order of approximated polynomial and the number of input of the network. It can also be expressed by the order of approximated polynomial accurately. Then, an algorithm to realize the approximation is given. Finally, two numerical examples are given for further illustrating the results. The obtained results are more important for constructing a feed-forward neural network with three-layer to approximate the class of polynomial functions and for realizing the approximation.