Study On Approximation Capability Of Neural Network And Spline Function

At present, the study of nonlinear science develops rapidly. With the development of computer technology and the efficient use of new mathematical analysis tools and methods, a series of important achievements and breakthroughs in nonlinear science have been accomplished. Theory of artificial neural network and spline functions are powerful tools to solve the nonlinear problem. It is often used in nonlinear dynamic system modeling and approaching. Meanwhile, it has been widely used in the field of automatic control and signal processing. In view of this, this paper first constructively studied the approximation and interpolation function of feedforward neural network with optimize activating function and fixed weights; then it introduces the application of recurrent neural network in nonlinear dynamic system modeling with a promotion and expansion of the recurrent neural network theory to approximate non-autonomous nonlinear dynamical system. Finally, by means of cubic spline interpolation function, a method to nonlinear dynamic system numerical solution is given. Meanwhile, the advantages and disadvantages of the method are analyzed. The error estimates are also given.Chapter II introduces three kinds of artificial neural networks:Feedforward Neural Networks, (FNNs), Continue Recurrent Neural Networks(CRNN), Discrete Recurrent Neural Networks (DRNN). Meanwhile, fundamental knowledge of nonlinear dynamical systems is studied. Chapter III constructively studied the approximation and interpolation function of feedforward neural network with optimize activating function and fixed weights. Chapter IV qualitatively studies the approximation ability of CRNN and DRNN to nonlinear dynamical systems. Firstly, it is proved that trajectory in a finite interval of any given non autonomous nonlinear dynamical system can be approximated by output neuron internal state of CRNN with any precision. Then extend and expand the results, dynamic CRNN can be used to approximate another non autonomous nonlinear dynamic system. Finally, by applying the CRNN discretization to nonlinear dynamic system modeling, using DRNN structure to approximate the nonlinear static analysis. Errors caused in the dynamic data sampling system and discrete modeling process are analyzed. Equivalent input and output structure to analyze and estimate error is excavated. The interdependent relations between approximation errors and model design errors are depicted and the method by choosing a suitable system structure to reduce the error is found. Chapter V quantitatively studies the numerical solution of the nonlinear dynamic system by cubic spline interpolation, which is called density research. Meanwhile, combined with characteristics of cubic spline interpolation function: convergence, stability and the second ordered smoothness, numerical solution of to construction integral nonlinear dynamic system is conducted, and errors are estimated.