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Semi-empirical Neural Network Modeling presents a new approach on how to quickly construct an accurate, multilayered neural network solution of differential equations. Current neural network methods have significant disadvantages, including a lengthy learning process and single-layered neural networks built on the finite element method (FEM). The strength of the new method presented in this book is the automatic inclusion of task parameters in the final solution formula, which eliminates the need for repeated problem-solving. This is especially important for constructing individual models with unique features. The book illustrates key concepts through a large number of specific problems, both hypothetical models and practical interest.
- Offers a new approach to neural networks using a unified simulation model at all stages of design and operation
- Illustrates this new approach with numerous concrete examples throughout the book
- Presents the methodology in separate and clearly-defined stages
Biomedical Engineers, researchers, and graduate students in neural networks and mathematical modeling
Chapter 1: Examples of problem statements and functionals
1.1 Problems for ordinary differential equations
1.2 Problems for partial differential equations for domains with fixed boundaries
1.3 Problems for partial differential equations in the case of the domain with variable borders
1.4 Inverse and other ill-posed tasks
Chapter 2: The choice of the functional basis (set of bases)
2.1 Multilayer perceptron
2.2 Networks with radial basis functions—RBF
2.3 Multilayer perceptron and RBF-networks with time delays
Chapter 3: Methods for the selection of parameters and structure of the neura network model
3.1 Structural algorithms
3.2 Methods of global non-linear optimization
3.3 Methods in the generalized definition
3.4 Methods of refinement of models of objects described by differential equations
Chapter 4: Results of computational experiments
4.1 Solving problems for ordinary differential equations
4.2 Solving problems for partial differential equations in domains with constant boundaries
4.3 Solving problems for partial differential equations for domains with variable boundaries
4.4 Solving inverse and other ill-posed problems
Chapter 5: Methods for constructing multilayer semi-empirical models
5.1 General description of methods
5.2 Application of methods for constructing approximate analytical solutions for ordinary differential equations
5.3 Application of multilayer methods for partial differential equations
5.4 Problems with real measurements
- No. of pages:
- © Academic Press 2020
- 22nd November 2019
- Academic Press
- Paperback ISBN:
- eBook ISBN:
Dmitry Tarkhov, born 14 Jan 1958 in St. Petersburg. In 1981 graduated with honors from the faculty of physics and mechanics of Leningrad Polytechnic Institute, majoring in Applied Mathematics and entered graduate school at the Department “Highermathematics”. After graduation, heworked at the Department as an assistant, then associate Professor and continues to work presently as a Professor. In 1987 he defended the thesis “the Straightening of the trajectories on the infinite dimensional torus”, for which he was awarded the degree of Ph.D. of physical and mathematical Sciences. In 1996, while working as a part-time chief systems analyst at the St. Petersburg Futures exchange he began studying neural networks. He has published more than 200 scientific papers on this topic. In 2006 he defended doctoral thesis “Mathematical modeling of technical objects on the basis of structural and parametrical adaptation of artificial neural networks”, for which he was awarded the degree of doctor of technical Sciences.
Professor, Section Head, Department of Higher Mathematics, Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russian Federation
Alexander Vasilyev was born in St. Petersburg (Leningrad) on 10 August 1948. After graduating mathematical school №239 with a gold medal and the Physics Faculty of Leningrad State University (LSU) with honors, he defended the Ph.D. thesis: New boundary value problems for ultrahyperbolic and wave equations, in the Leningrad branch of the Steklov Mathematical Institute in 1978. Working since 1980 at the Department of higher mathematics of Peter the Great St. Petersburg Polytechnic University as an Associate Professor and since 2007 as a Professor, he read advanced courses and electives in various areas of modern mathematics, led seminars.He prepared and in 2011 defended his doctoral thesis: Mathematical modeling of systems with distributed parameters based on neural network technique, for the degree of Doctor of Technical Sciences in the specialty 05.13.18 – “Mathematical modeling, numerical methods, and software.” Professor Vasilyev’s scientific interests are in the field of differential equations, mathematical physics, ill-posed problems, meshless methods, neuro-mathematics, neural network modeling of complex systems with approximately specified characteristics, heterogeneous data, digital twins, deep learning, global optimization, evolutionary algorithms, big transport system, environmental problems, educational projects.He is the author and co-author of three monographs, chapters in the reference book, chapters in two collective monographs, textbook (with the Ministry of Education stamp) – in Russian; he published about 180 works devoted to neural network modeling; he has Honors Diploma of the Ministry of education of the Russian Federation, the Diploma and Awards of Polytechnic University Board. Professor Vasilyev is the Chairman, the member of the Organizing Committee of the conferences; he is the head, the central executive and participant of projects supported by grants of RF, a member of the editorial board of the “Journal Mathematical Modeling and Geometry.” He is fond of painting and graphics. The book "Semi-empirical neural network modeling" (with co-authors) is the first monograph in English.
Professor, Peter the Great St. Petersburg Polytechnic University
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