Theoretical Solutions to Several Open Problems in Neural
Network Research: A Review of Some Recent Work

Zongben Xu

Xi'an Jiaotong University, Xi'an, China

Abstract
    We review the recent work done by our group on the theoretical analysis of artificial neural networks. We focus on answering the two questions in feedforward networks (FNNs): (i) how to measure the essential approximation capability of a FNN, and (ii) what conditions guarantee the convergence of on-line BP training procedure. In feedback neural networks, we provide solutions for the following three questions: (i) what are the necessary and sufficient conditions for global convergence/stability of the Hopfield-type neural networks, (ii) how to distinguish global exponential stability (GES) from global asymptotic stability (GAS), and (iii) whether there is any difference to model a neural network in terms of its local field state (internal state) and in terms of its neuron state (external state). In each case, we summarize the state-of-the-art results and explain their impact and significance.

Biosketch 
    Zong-Ben Xu received the MS degree in mathematics in 1981 and the PhD degree in applied mathematics in 1987 from Xi`an Jiaotong University, China. In 1998, he was a postdoctoral researcher in the Department of Mathematics, the University of Strathclyde, United Kingdom. He worked as a research fellow at Chinese University of Hong Kong and Hong Kong Polytechnic University from 1996 to 2000. In 2000, he worked as visiting professor in the University of Napoli Federico II (Italy) and in 2001, in the University of Essex (UK). He has been with the Faculty of Science and Institute for Information and System Sciences at Xi`an Jiaotong University since 1982, where he was promoted to associate professor in 1987 and full professor in 1991, and now serves as an authorized PhD supervisor in mathematics and computer science, Vice President of Xi’an Jiaotong University, and Director of the Institute for Information and System Sciences.
    Professor Xu currently makes several important services for government and professional societies, including Consultant expert for National (973) Program in Key Basic Science Research and Development (Information group), Ministry of Science and Technology of China; Evaluation Committee member for mathematics degree, Academic Degree Commission of the Chinese Council; Committee member in Scientific Committee of Education Ministry of China (Mathematics and Physics Group); Vice-Director of the Teaching Guidance Committee for Mathematics and Statistics Majors, the Education Ministry of China; Member in the Expert Evaluation Committee for Natural Science Foundation of China (Computer Science Group), The National Committee for Natural Science Foundation of China; Vice-president of Computational Intelligence Society of China; Editor-in-chief of the series of textbooks on information and computational sciences, Higher Education Press of China; Co-editor of nine national and international journals.
    Professor Xu has published 116 academic papers, mostly on well-known journals ， such as IEEE Trans. on Evolutionary Computation, on Pattern Analysis and Machine Intelligence, on Neural Networks and on System, Man & Cybernetics, Proc. Royal Soc. London,, SIAM J Numerical Analysis, Science of China, etc. His current interest is in computational Intelligence (particularly, neural networks, evolutionary computation, data mining theory and algorithms), Nonlinear Functional Analysis and Geometry of Banach Spaces.