基于Bernstein多项式的SISO三层前向神经网络的设计与逼近
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摘要
利用一元Bernstein多项式在相邻等距剖分点的差值和Sigmodial转移函数性质设计单输入单输出(single input single output,SISO)三层前向神经网络,并给出选取连接权和阈值的方法。此外,依据一元Bernstein多项式逼近连续函数定理证明SISO三层前向神经网络对连续函数也具有逼近性,进而通过实例获得该网络的一种输入输出解析表达式。
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.
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.
引文
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[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-forw ard neural netw ork based on the layer-bylayer 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 netw orks w ith single hidden layer and the best polynomial approximation[J]. Acta M athematica 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 netw ork 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 netw ork 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 netw orks[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 M ISO polygonal fuzzy neural netw ork[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. M ultivariate function approximation[M]. Beijing:Science Press,1988.