Semi-empirical Neural Network Modeling and Digital Twins Development - 1st Edition - ISBN: 9780128156513

Semi-empirical Neural Network Modeling and Digital Twins Development

1st Edition

Authors: Dmitriy Tarkhov Alexander Nikolayevich Vasilyev
Paperback ISBN: 9780128156513
Imprint: Academic Press
Published Date: 25th November 2019
Page Count: 284
Sales tax will be calculated at check-out Price includes VAT/GST

Institutional Subscription

Secure Checkout

Personal information is secured with SSL technology.

Free Shipping

Free global shipping
No minimum order.


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.

Key Features

  • 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

Table of Contents


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
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
5.4 Problems with real measurements


No. of pages:
© Academic Press 2020
25th November 2019
Academic Press
Paperback ISBN:

About the Author

Dmitriy Tarkhov

Dr. D.A. Tarkhov is a professor in the Department of Higher Mathematics at Peter the Great St. Petersburg Polytechnic University. Dr. Tarkhov has also worked as a chief systems analyst at the St. Petersburg Futures Exchange, where he began studying neural networks. He has published more than 150 scientific papers on neural networks and the associated mathematics, including “Parametric Neural Network Modeling in Engineering” and “Multilayer Neural Network Models Based on Grid Methods.” Dr. Tarkhov is a leading researcher in the development and application of neural networks across a wide variety of biomedical and scientific applications, including air pollution, migration stream modeling, metallurgy, materials rupture, deformation, and destruction modeling.

Affiliations and Expertise

Professor, Section Head, Department of Higher Mathematics, Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russian Federation

Alexander Nikolayevich Vasilyev

Alexander Vasilyev was born in St. Petersburg (Leningrad) 10 August 1948. Graduated in mathematical school №239 with a gold medal and continued his studies at the Physics Faculty of Leningrad State University (LSU), which he graduated with honors in "mathematical physics.Has post-graduate studies at the Research Institute of Physics of LSU via presentation of the thesis for the degree of candidate of Physical and Mathematical Sciences (Ph.D.) in the specialty 01.01.02 –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 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. Scientific and pedagogical experience of 47 years. 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.

Affiliations and Expertise

Professor, Peter the Great St. Petersburg Polytechnic University

Ratings and Reviews