Interpolation neural network constructed by the step path and its approximation performance
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Interpolation neural network constructed by the step path and its approximation performance Guijun Wang1 • Weiming Xiao1 • Yujie Tao2 Received: 2 June 2020 / Revised: 8 October 2020 / Accepted: 13 October 2020 Ó Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Traditional neural network is a nonlinear dynamic system formed by a large number of neurons connected to each other, it can be widely used in many research fields, such as data mining, system identification and intelligent control. The neural networks can not only deal with data problems through the unique thinking of the human brain, but can also solve the multi input and multi output problem which is difficult to be completed by the traditional computer. In this paper, in the sense of an equidistance dissection of two-dimensional input space, some new connection weights and thresholds are determined by the interpolations and arithmetic mean values of some data pairs at the adjacent intersecting points, respectively, and a new forward interpolation neural network is constructed by a step path of two-dimensional dissection. Secondly, it is proved that the interpolation neural network can approximate to a continuous function by using characteristic properties of a Sigmoidal activation function. Finally, the approximation performance of the constructed network is tested in the light of the t- hypothesis test method in statistical inference, and the approximation errors of the interpolation network are compared with that of another polynomial neural network. Keywords Step path Activation function Interpolation neural network Approximation Hypothesis test
1 Introduction As we all know, an neural network is an important intelligent control technology in the field of artificial intelligence. It not only has the function of logic reasoning and numerical calculation, but has also the ability to approximate the nonlinear function. Actually, an artificial neural network is suitable for modeling complex nonlinear system problems by simulating human brain intelligence and memory, and have many advantages such as distributed storage, parallel processing, autonomous learning and selforganization. In recent years, the neural networks have penetrated into various fields of research, especially in the fields of intelligent control, pattern recognition, signal processing, sensing technology and robot, and so on. & Yujie Tao [email protected] 1
School of Mathematical Science, Tianjin Normal University, Tianjin 300387, China
2
School of Mathematics, Tonghua Normal University, Tonghua, Jilin 134002, China
In 2003, Cao and Xu et al. constructed a three-layer forward neural network only related to a polynomial order by selecting appropriate activation function in [1]. The network can approximate the polynomial with arbitrary accuracy according to specific algorithm, where the total number of neurons in the hidden layer of the network n(number of nodes) is only related to the order r of the polynomi
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