中文摘要：

首先用构造性的方法证明：对于任意的n阶多元多项式函数，可以构造一个三层前向神经网络以任意精度逼近该多项式，所构造网络的隐层节点个数仅与多项式的维数d和阶数n有关．然后，我们给出实现这一逼近的具体算法．最后，给出两个算例进一步验证所得的理论结果．本文结果对神经网络逼近多元多项式函数的具体网络构造以及实现这一逼近的方法等问题具有指导意义．

英文摘要：

It is investigated that multivariate polynomial functions with n order are approximated by feedforward neural networks with three layers. Firstly, for a given polynomial function with n order, a feedforward neural network with three layers is designed by a constructive method to approximate the polynomial with any degree of accuracy. The number of hidden layer nodes of. the constructed network only depends on the order and dimension of approximated polynomial. Then, an algorithm to realize the approximation is given. Finally, two numerical examples are given for further illustration of the results. The obtained results have a guidance signification to construct feedforward neural networks with three layers to approximate the class of polynomial functions and realize the approximation.