(The) approximation and analysis of the function using neural network

Abstract

The Neural Network has been used in various areas since its development of 1950````s; pattern recognition, quality control, etc. Besides those uses, we can also use the neural network to find an estimated value of a function. The neural network has the learning ability. The neural network can learn the relationship between its input and output. We can use this ability to find an estimated value of a function. In other words, we can predict the output of a function with a properly trained neural network. We show the neural network both the input and the corresponding output of a function. After training, we give the neural network an arbitrary input, and the neural network will give us the corresponding output. This output is an ````estimated```` output of a function because it is obtained through the neural network that is trained to mimic the behavior of the function. We can use this method for an unknown function about which we know of some inputs and corresponding outputs. Also, we can use the neural network not only to mimic the behavior of a function but also to analyze it. During the training, the neural network learns about the characteristic properties of the given function, and acts upon that characteristic instead of exactly copying the behavior of the function. In other words, the neural network may behave quite differently from the given function, but its behavior shows a certain characteristic of the given function. In this paper, I will present a brief introduction of the neural network, and the actual use of the neural network for the prediction of a functional output and the analysis of the function, especially about the logistic map. During the discussion, I will present two views concerning the work of the neural network. One is to view it from the computational point, and another is to view it as a classification process.