Study On Interpolation And Approximation Problem Of Neural Networks

Theory of artificial neural network is powerful tool to solve the nonlinear problems and more and more scholars attention to it.The interpolation of neural network and the approximation ability to nonlinear dynamica system are one of the hot and difficult aspect in theory research of artificial neural network.Besides,compare to the learning networks obtained by all kinds of algorithms based on sample learning,the networks obtained by construction method overcome the local limit value of learning networks and slow speed of convergence and so on.Moreover,because of recurrent neural networks with high nonlinear characteristics,so it is a good way to describe the characteristics of nonlinear dynamic system.In view of this,in this paper,we discuss the interpolation and approximation of feedforward neural networks and approximate nonlinear dynamical system by recurrent neural networks.Chapter II introduces two kinds of artificial neural networks:Feedforward Neural Networks(FNN)and Recurrent Neural Networks(RNN).At the same time,basic knowledge of nonlinear dynamic system,approximation of neural networks and interpolation about neural networks are introduced.Chapter III adopts the construction method studied the approximation and interpolation of feedforward neural network.when the activation function is Sigmoidal function,we can construct a one hidden layer feedforward neural networks.Under certain conditions,the precise interpolation network of one variable is exist and construct the approximate interpolation neural network.Besides,we give the error between exact interpolation neural network and approximate interpolation neural network.For multivariate functions,Using the method in one-dimensional space,through the inner product for interpolation nodes,we construct a multivariate precise interpolation neural network and an approximate interpolation neural network.Furthermore,we also give the error of the multivariate precise interpolation neural network and multivariate approximate interpolation neural network.Chapter IV shows that the family of recurrent neural networks with saturated linear transfer functions and weight matrices of rank 1 are essentially equivalent to feedforward neural networks with recurrent layers.Therefore,they inherit the universal approximation property of real-valued functions in on variable in a stronger sense.Chapter V of this paper tries to summarize and instruduces the further research work in the future.