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 hidden-layer 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.