JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2018, Vol. 53 ›› Issue (9): 55-61.doi: 10.6040/j.issn.1671-9352.0.2017.606

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Design and approximation of SISO three layers feedforward neural network based on Bernstein polynomials

XIAO Wei-ming, WANG Gui-jun*   

  1. School of Mathematical Science, Tianjin Normal University, Tianjin 300387, China
  • Received:2017-11-24 Online:2018-09-20 Published:2018-09-10

Abstract: A single input single output(SISO)three layers feedforward neural network was designed by using the difference value between adjacent equidistant subdivision points of unary Bernstein polynomial with a Sigmodial transfer function, and a method of selecting the connection weights and thresholds was given. In addition, according to the approximation theorem for unary Bernstein polynomial, we proved that SISO three layers feedforward neural network could also approximate a continuous function. The analytical expression of the neural network was obtained by an example.

Key words: equidistant subdivision, Bernstein polynomials, Sigmodial transfer function, feedforward neural network, approximation

CLC Number: 

  • TP183
[1] RUMELHART D E, HINTON C E, WILIAMS R J. Learning representations by back-propagating errors[J]. Nature, 1986, 323(9):533-536.
[2] CHEN Tianping, CHEN Hong, LIU Ruey-wen. Approximation capability in C(Rn)by multilayer feed-forward networks and related problems[J]. IEEE Transactions on Networks, 1995, 6(1):25-30.
[3] SCARSELLI F, TSOI A C. Universal approximation using feed-forward neural network: a survey of some existing methods, and some new results[J]. Neural Networks, 1998, 11(1):15-37.
[4] 田传俊,韦岗.前向神经网络的一种快速分层线性优化算法[J].电子学报,2001,29(11): 1495-1498. TIAN Chuanjun, WEI Gang. A new layer-wise linearized algorithm for feed-forward neural network[J]. Acta Electronica Sinica, 2001, 29(11): 1495-1498.
[5] 谢宏,程浩忠,牛东晓. 前向神经网络的神经元分层逐个线性优化快速学习算法[J].电子学报,2005,33(1):111-114. XIE Hong, CHEN Haozhong, NIU Dongxiao. A fast learning algorithm for feed-forward neural network based on the layer-by-layer and neuron-by-neuron optimizing procedure[J]. Acta Electronica Sinica, 2005, 33(1):111-114.
[6] 曹飞龙,张永全,张卫国.单隐层神经网络与最佳多项式逼近[J]. 数学学报,2007, 50(2):387-392. CAO Feilong, ZHANG Yongquan, ZHANG Weiguo. Neural networks with single hidden layer and the best polynomial approximation[J]. Acta Mathematica Sinica, 2007, 50(2):387-392.
[7] 刘普寅.模糊神经网络及其应用[D]. 北京:北京师范大学,2002. LIU Puyin. Fuzzy neural network and its application[D]. Beijing: Beijing Normal University, 2002.
[8] 王贵君,李晓萍. K-积分模意义下折线模糊神经网络的泛逼近性[J]. 中国科学:信息科学,2012,42(3):362-378. WANG Guijun, LI Xiaoping. Universal approximation of polygonal fuzzy neural network science of K-integral norms[J]. Scientia Sinica Informations, 2012, 42(3):362-378.
[9] 何英,王贵君.折线模糊神经网络的共轭梯度算法[J].电子学报,2012,40(10):2079-2084. HE Ying, WANG Guijun. Conjugate gradient algorithm of the polygonal fuzzy neural networks[J]. Acta Electronica Sinica, 2012, 40(10):2079-2084.
[10] 王贵君,何英,李晓萍.基于MISO折线模糊神经网络的优化算法[J].中国科学:信息科学, 2015, 45(5):650-667. WANG Guijun, HE Ying, LI Xiaoping. Optimization algorithm for MISO polygonal fuzzy neural network[J]. Scientia Sinica Informations, 2015, 45(5):650-667.
[11] 佩捷,施雨辰.伯恩斯坦多项式与贝齐尔曲面[M].哈尔滨:哈尔滨工业大学出版社, 2013. PEI Jie, SHI Yuchen. Bernstein polynomials and Bezil surfaces[M]. Harbin: Harbin Institute of Technology Press, 2013.
[12] 王仁宏,梁学章.多元函数逼近[M].北京:科学出版社, 1988. WANG Renhong, LIANG Xuezhang. Multivariate function approximation[M]. Beijing: Science Press, 1988.
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