Mathematical Methods for Neural Network Analysis and Design

Basic Approach and Philosophy

This book is biased towards a particular viewpoint regarding what mathematical methodologies are most important for ANN system analysis and design. In particular, this book emphasizes: (i) nonlinear deterministic and stochastic dynamical systems theory, (ii) nonlinear optimization theory, and (iii) statistical inference theory approaches to ANN system analysis and design. Other branches of mathematics have been deliberately omitted in order to force the reader to focus upon fundamental issues.

The methods of nonlinear analysis described in this book are directly applicable to large-scale complex systems. The philosophy of this book is to make weak statements about complex systems as opposed to strong statements about simplified systems. An original contribution of this book is the presentation of ANN algorithms as special cases of the classical engineering and mathematics literature in such a way as to teach and develop intuitions about ANN system analysis and design. Although this textbook emphasizes classical well-established theorems in mathematics and engineering relevant to ANN system analysis and design, the theorems and their proofs have been considerably rewritten in order to: (i) develop a uniformity of notation throughout this textbook, and (ii) to simplify the mathematical presentation.

Furthermore, unlike other books concerned with neural network analysis and design, the details of specific neural network architectures are considered primarily in the problems and examples rather than in the main text. The reason for this unique approach is to clarify how various methods of neural network analysis and design can be examined within a unified framework which is based firmly in traditional engineering design. Unlike most other books in the field of ANN systems, this book is not organized by algorithms. The neural network field has exploded rapidly and there is a tremendously wide variety of algorithms which have been introduced and are being introduced on an annual basis. By presenting fundamental mathematical tools which are applicable to large classes of high-dimensional nonlinear information processing systems, it is hoped that the researcher can use this book to approach the analysis of most novel linear and nonlinear ANN systems which have already been developed and which will also be developed in the future.

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