Aspects of the numerical analysis of neural networks

Abstract

This article starts with a brief introduction to neural networks for those unfamiliar with the basic concepts, together with a very brief overview of mathematical approaches to the subject. This is followed by a more detailed look at three areas of research which are of particular interest to numerical analysts.The first area is approximation theory. IfKis a compact set in ℝn, for somen, then it is proved that a semilinear feedforward network with one hidden layer can uniformly approximate any continuous function inC(K) to any required accuracy. A discussion of known results and open questions on the degree of approximation is included. We also consider the relevance of radial basis functions to neural networks.The second area considered is that of learning algorithms. A detailed analysis of one popular algorithm (the delta rule) will be given, indicating why one implementation leads to a stable numerical process, whereas an initially attractive variant (essentially a form of steepest descent) does not. Similar considerations apply to the backpropagation algorithm. The effect of filtering and other preprocessing of the input data will also be discussed systematically.Finally some applications of neural networks to numerical computation are considered.