前馈神经网络在多元函数逼近中的应用
Application of Feedforward Neura Networks in Multivariate Function Approximation
DOI: 10.12677/SA.2020.96110, PDF,   
作者: 葛悠然, 王汉权*：云南财经大学，统计与数学学院，云南 昆明；翟九媛：云南大学，经济学院，云南 昆明；马尧鹏：云南财经大学，云南 昆明
关键词: 前馈神经网络；函数逼近；隐含层节点数目；Feedforward Neural Network； Function Approximation； Number of Hidden Layer Nodes
摘要: 给定一组数据(例如一些点及相应点处的函数值)，找到未知函数的表达公式——函数逼近问题是数学与工程应用中的一个基本问题。传统的数值方法多采用多项式插值法(例如拉格朗日插值法、牛顿插值法、三次样条法等)，本文通过构造前馈神经网络函数得到未知函数的表达式，讨论其处理函数逼近问题的优缺点。具体说来，先介绍训练多元函数的前馈神经网络的详细计算过程，然后分析隐含层节点数目对该网络的精度影响问题。最后通过数值计算结果证实前馈神经网络可用来逼近一元函数、二元函数、三元函数，能够达到较高的计算精度。本文的讨论适用于其他类人工神经网络在四元或四元以上的多元函数逼近问题的研究，也有助于理解相关人工神经网络的基本性质与作用。
Abstract: Given a set of data (such as some points and function values at corresponding points), finding the expression formula of the unknown function—the function approximation problem is a fundamental problem in mathematics and engineering applications. Traditional numerical methods mostly use polynomial interpolation (such as Lagrangian interpolation, Newton interpolation, cubic spline method, etc.). In this paper, the expression of the unknown function is obtained by constructing the feedforward neural network function, and the advantages and disadvantages of the approximation problem of the processing function are discussed. Specifically, the detailed calculation process of the feedforward neural network for training multivariate functions is first introduced, and then the influence of the number of hidden layer nodes on the accuracy of the network is analyzed. Finally, the numerical results show that the feedforward neural network can be used to approximate the unary function, the binary function and the ternary function, which can achieve higher calculation accuracy. The discussion in this paper is applicable to the study of multivariate function approximation problems of other artificial neural networks in quaternary or quadruple, and it also helps to understand the basic properties and effects of related artificial neural networks.
文章引用：葛悠然, 翟九媛, 马尧鹏, 王汉权. 前馈神经网络在多元函数逼近中的应用[J]. 统计学与应用, 2020, 9(6): 1048-1059. https://doi.org/10.12677/SA.2020.96110

参考文献

[1] 沈燮昌. 逼近论发展史简述(一) [J]. 数学研究及应用, 1982, 2(2): 171-180.
[2] 徐学良. 人工神经网络的发展及现状[J]. 微电子学, 2017, 47(2): 239-242.
[3] Bulsari, A. (1993) Some Analytical Solutions to the General Approximation Problem for Feedforward Neural Networks. Neural Networks, 6, 991-996.
[Google Scholar] [CrossRef] 
[4] Suzuki, S. (1998) Constructive Function-Approximation by Three-Layer Artificial Neural Networks. Neural Networks, 11, 1049-1058.
[Google Scholar] [CrossRef] 
[5] Twomey, J.M. and Smith, A.E. (1998) Bias and Variance of Validation Methods for Function Approximation Neural Networks under Conditions of Sparse Data. IEEE Transactions on Systems, Man and Cybernetics, Part C (Applications and Reviews), 28, 417-430.
[Google Scholar] [CrossRef] 
[6] Ferrari, S. and Stengel, R.F. (2005) Smooth Function Approximation Using Neural Networks. IEEE Transactions on Neural Networks, 16, 24-38.
[Google Scholar] [CrossRef] 
[7] Yao, S., Wei, C.J. and He, Z.Y. (2013) Evolving Wavelet Neural Networks for Function Approximation. Electronics Letters, 17, 586-594.
[8] 韦岗, 李华, 徐秉铮. 关于前馈多层神经网络多维函数逼近能力的一个定理[J]. 电子与信息学报, 1997, 19(4): 433-438.
[9] 王强, 余岳峰, 张浩炯. 利用人工神经网络实现函数逼近[J]. 计算机仿真, 2002(5): 44-47.
[10] 侯木舟. 基于构造型前馈神经网络的函数逼近与应用[D]: [博士学位论文]. 长沙: 中南大学, 2009.
[11] 李鹏柱. 关于神经网络与样条函数的逼近性能研究[D]: [硕士学位论文]. 银川: 宁夏大学, 2016.
[12] 许洋. 前馈型神经网络算法优化分析[J]. 硅谷, 2014(13): 66, 42.
[13] Heaton, J.B., Polson, N.G. and Witte, J.H. (2018) Deep Learning in Finance.
[14] 周开利. 神经网络模型及其MATLAB仿真程序设计[M]. 北京: 清华大学出版社, 2005.
[15] 孙永生. 函数逼近论[M]. 北京: 北京师范大学出版社, 1989.
[16] 李晓东, 胡志恒, 虞厥邦. 一种前馈神经网络的快速学习算法[J]. 信号处理, 2004, 20(2): 184-187.
[17] Foresee, F.D. and Hagan, M.T. (1997) Gauss-Newton Approximation to Bayesian Regularization. International-Joint Conference on Neural Network, 1930-1935.

为你推荐