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This book introduces a variety of neural network methods for solving differential equations arising in science and engineering. The emphasis is placed on a deep understanding of the neural network techniques, which has been presented in a mostly heuristic and intuitive manner. This approach will enable the reader to understand the working, efficiency and shortcomings of each neural network technique for solving differential equations. The objective of this book is to provide the reader with a sound understanding of the foundations of neural networks and a comprehensive introduction to neural network methods for solving differential equations together with recent developments in the techniques and their applications.

The book comprises four major sections. Section I consists of a brief overview of differential equations and the relevant physical problems arising in science and engineering. Section II illustrates the history of neural networks starting from their beginnings in the 1940s through to the renewed interest of the 1980s. A general introduction to neural networks and learning technologies is presented in Section III. This section also includes the description of the multilayer perceptron and its learning methods. In Section IV, the different neural network methods for solving differential equations are introduced, including discussion of the most recent developments in the field.

Advanced students and researchers in mathematics, computer science and various disciplines in science and engineering will find this book a valuable reference source.

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Preface. Introduction.

1 Overview of Differential Equation. 1.1 Classification of Differential Equations. 1.2 Types of Differential Equation Problems. 1.3 Differential Equations Associated with Physical Problems Arising in Engineering. 1.4 General Introduction of Numerical Methods for Solving Differential Equations. 1.5 Advantages of Neural Network Method for Solving Differential Equations.

2 History of Neural Networks. 2.1 The 1940´s: The Beginning of Neural Nets. 2.2 The 1950´s and 1960´s: The First Golden Age of Neural Networks. 2.3 The 1970´s: The Quiet Years. 2.4 The 1980´s: Renewed Enthusiasm.

3 Preliminaries of Neural Networks. 3.1 What is Neural Network? 3.2 Biological Neural Network. 3.3 Artificial Neural Network. 3.4 Mathematical Model of Artificial Neural Network. 3.5 Activation Function. 3.6 Neural Network Architecture. 3.7 Learning in Neural Networks. 3.8 Multi-layer Perceptron. 3.9 Neural Networks as Universal Approximator.

4 Neural Network Methods for Solving Differential Equations. 4.1 Method of Multilayer Perceptron Neural Network. 4.2 Method of Radial Basis Function Neural Networks. 4.3 Method of Multiquadric Radial Basis Function Neural Network. 4.4 Method of Cellular Neural Networks. 4.5 Method of Finite Element Neural Networks. 4.6 Method of Wavelet Neural Networks. 4.7 Some Workout Examples.

Conclusion. Appendix. References. Index.

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"The book is intended to enable the reader to get an image on the variety of NN and the NN methods can be used in solving differential equations. It is a valuable reference material both from the presentation point of view and the provided references." (Liviu Goras, zbMATH 1328.92006, 2016)

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