Chinese Journal of Computers   Full Text
Title	Approximation Method of Multivariate Polynomials by Feedforward Neural Networks
Authors	WANG Jian-Jun1),2) XU Zong-Ben2)
Address	1)(School of Mathematics & Statistics, Southwest University, Chongqing 400715) 2)(Institute for Information and System Science, Xi’an Jiaotong University, Xi’an 710049)
Year	2009
Issue	No.12(2482—2488)
Abstract &   Background	Abstract 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. Keywords feedforward neural network; multivariate polynomials; approximation; algorithm Background Artificial neural networks have been extensively applied in various fields of science and engineering. Various problems concerning application of neural networks in science and engineering can be converted into problems of approximating functions by superposition of the neural activation functions of the networks. The approximation of multivariate functions by the FNNs has been widely studied in past years, with various significant results, concerning density or complexity. However, from the respective of application, the algorithm research on approximation of the neural networks is more hopeful, especially, the determination of topology of neural networks and the parameter of algorithm. Polynomial is a class of fundamental functions. There are many methods to realize its approximation. The authors study an algorithm based on the theory of neural networks, and give theory and experiments of the algorithm. These results reveal the relationship between topology of networks and approximation accuracy in approximation of polynomial. This research is supported by the National Basic Research Program(973 Program) of China (No.2007CB311000) and the National Natural Science Foundation of China (Nos.10726040,70531030,10701062,10826081), the Key Project of Chinese Ministry of Education (No.108176), China Postdoctoral Science Foundation (No.20080431237).The group already achieved some research works in the area of theory and application of neural works, which was published in area premium journals such as Science in China Series E: Information Science, Neural Networks, Chinese Journal of Computers and so on. As an essential part of the project, both the theory and experimental results will deepen the research and contribute to above projects. Currently, the authors and their research group are conducting intensive research in this field to get more and better results.