An Introduction to Neural Network Methods for Differential Equations

ISBN-13: 9789401798150 / Angielski / Miękka / 2015 / 114 str.

Neha Yadav; Anupam Yadav; Manoj Kumar

Opis

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."

Recenzje

"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)

Spis treści

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.

O autorze

Dr. Neha Yadav, Assistant Professor (Mathematics), Department of Applied Science, ITM University Gurgaon, Haryana-122017, India. Specialization: Numerical Analysis and Soft Computing Techniques, Differential Equations, Boundary Value Problems. Total Experience: 03 Years Teaching and 04 years Research Experience. Research Papers in Refereed SCI journals : 03 (Published), 03 (Submitted). Awards and Prizes: (i) Travel Award from CSIR-HRDG and NBHM (Govt. of India) to visit University of Strathclyde, Glasgow, U.K. in the year 2013. (ii) Qualified UGC-NET JRF in the year 2010. (iii) Selected for half financial to participate in “School and Conference on Computation Methods in Dynamics” at Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, from 20 June to 8 July 2011. (iv) Selected for MHRD Institute Fellowship in PhD at MNNIT Allahabad. (v) Selected for Summer Research Fellowship Programme jointly sponsored by IASc (Bangalore), INSA(New Delhi) and NASI(Allahabad).

Dr. Anupam Yadav, Assistant Professor (Mathematics). National Institute of Technology Uttarakhand. Pauri Garhwal, Uttarakhand - 246174. Specialization: Soft Computing Techniques, Swarm Intelligence, Artificial Intelligence. Area of Research: Optimization, Operations Research. Research Papers in Refereed SCI journals : 04 (Published), 04 (Submitted). Awards: Award from NBHM-DAE (Govt. of India) to visit Glasgow, U. K. in the year 2013. Award from CSIR-HRDG (Govt. Of India) to visit Taipei, Taiwan in the year 2011. CSIR – JRF (Mathematical Sciences) in the year 2009. GATE – 2009 with All India Rank 95. Positions held: Asst. Professor National Institute of Technology Uttarakhand, India. Research Professor: DPST Center, Korea University, Seoul, South Korea. Senior Research Fellow: IIT Roorkee, India. Junior Research Fellow: IIT Roorkee, India.

Dr. Manoj Kumar, Associate Professor (Mathematics), Motilal Nehru National Institute of Technology, Allahabad, India-211004. Specializations: Numerical Analysis and Computer Application, Simulation & Modeling. Area of Research: Numerical Analysis/Operation Research/Mathematical Modeling/Partial Differential Equations/ Computational Fluid Dynamics. Teaching Experience : Since 2001 teaching B.Tech, M.Tech, MCA classes and guiding PhD/ Post-Doctoral Students. Research Papers in Refereed SCI Journals:  67. PhD Student Guided: 09 (Awarded) , 02(Work in Progress). Post-Doctoral Guidance:04. Independent Research Grants: 04. Reviewer of International Journals: 11.