This book presents a comprehensive exploration of the dynamical system approach in numerical linear algebra, with a special focus on computing generalized inverses, solving systems of linear equations, and addressing linear matrix equations. Bridging four major scientific domains numerical linear algebra, recurrent neural networks (RNNs), dynamical systems, and unconstrained nonlinear optimization this book offers a unique, interdisciplinary perspective.

Generalized Matrix Inversion: A Machine Learning Approach explores the theory and application of recurrent neural networks, particularly continuous-time recurrent neural networks (CTRNNs), which use systems of ordinary differential equations to model the influence of inputs on neurons. Special attention is given to CTRNNs designed for finding zeros of equations or minimizing nonlinear functions, with detailed coverage of two important classes: Gradient Neural Networks (GNN) and Zhang (Zeroing) Neural Networks (ZNN). Both time-varying and time-invariant models are examined across scalar, vector, and matrix cases.

Based on the authors research that has been published in leading scientific journals, the book spans a variety of disciplines, including linear and multilinear algebra, generalized inverses, recurrent neural networks, dynamical systems, time-varying problem solving, and unconstrained nonlinear optimization. Readers will find a global overview of activation functions, rigorous convergence analysis, and innovative improvements in the definition of error functions for GNN and ZNN dynamic systems.

Generalized Matrix Inversion: A Machine Learning Approach is an essential resource for researchers and practitioners seeking advanced methods at the intersection of machine learning, optimization, and matrix computation.

Autorentext

Predrag S. Stanimirovic received his Ph.D. in Computer Science at University of Nis, Serbia. He is full Professor at University of Nis, Faculty of Sciences and Mathematics, Department of Computer Science, Nis, Serbia. He has 36 years of experience in scientific research in diverse fields of mathematics and computer science, spanning multiple branches of numerical linear algebra, recurrent neural networks, linear algebra, nonlinear optimization, symbolic computation and others. His main research topics include Numerical Linear Algebra, Operations Research, Recurrent Neural Networks and Symbolic Computation. He has published over 350 publications in scientific journals, including 7 research monographs, 6 text-books, and over 80 peer-reviewed research articles published in conference proceedings and book chapters. He is an editorial board member of more than 20 scientific journals, 5 of which belong to Journal Citation Report (JCR) list. Currently he is section editor of the journals Electronic Research Archive (ERA), Filomat, Journal of Mathematics, Contemporary Mathematics (CM), Facta Universitatis, Series: Mathematics and Informatics, and several other journals. He is an author in the World Rank List of 2% best authors in 2021, 2022 and 2023.

Yimin Wei received his Ph.D in Computational Mathematics at Fudan University. He is a full Professor with the School of Mathematical Sciences, Fudan University. He is the author of more than 200 technical journal articles and five monographs published by Elsevier, Springer, World Scientific, EDP Science, and Science Press. His current research interests include multilinear algebra and numerical linear algebra with their applications. He is an author in the World's Top 2% Scientists in 2021, 2022 and 2023

Shuai Li received the M.E. degree in automatic control engineering from University of Science and Technology of China, China, and a Ph.D. degree in Electrical and Computer Engineering from Stevens Institute of Technology, Hoboken, NJ, USA in 2014. He is currently a full professor with Faculty of Information Technology and Electrical Engineering, University of Oulu, Finland and an adjunct professor with VTT (Technical Research Center of Finland), Oulu, Finland. His current research interests include dynamic neural networks, robotics, machine learning, and autonomous systems.

Dimitrios K. Gerontitis received a B.S. degree in Mathematics and the M.S. degree in Theoretical Informatics and Systems and Control Theory from the Aristotle University of Thessaloniki, Thessaloniki, Greece, in 2013 and 2016, respectively. He is currently pursuing a Ph.D. degree in the development of intelligent computational methods for solving time-varying problems at the Department of Information and Electronic Engineering, International Hellenic University (IHU). His main research interests include neural networks, optimization methods, robotics, and numerical linear algebra.

Xinwei Cao is a full professor at the School of Business, Jiangnan University, China, with a distinguished interdisciplinary background in both management and computing. She earned her PhD through a joint program between the School of Management at Fudan University and the School of Business at the Chinese University of Hong Kong. Over the years, Dr. Cao has actively collaborated with experts in computing and artificial intelligence to seamlessly integrate AI into management practices. Dr. Cao has published over 50 peer-reviewed scientific papers, reflecting her significant contributions to academia. In addition to her academic work she serves as a consultant and an independent director on audit committees for listed companies.

Inhalt

1. Background Information.- 2 Gradient Neural Network (GNN) and their Modifications.- 3 Zeroing Neural Network (ZNN).- 4 From iterations to ZNNs and vice versa, 5 Modified ZNN dynamical systems.