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: 1st November 2019
Page Count: 320
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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.

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 statements of problems and functions
1.1. Problems for Ordinary Differential Equations
1.1.1 Porous Catalyst
1.1.2 Chemical Reactor
1.1.3 Stiff Differential Equation
1.1.4 Differential-Algebraic Problem
1.2. Problems for Partial Differential Equations for Domains with Fixed Boundaries
1.2.1 The Laplace Equation on Plane (Including L-Region) and in Space
1.2.2 The Poisson Problem
1.2.3 The Schrodinger Equation with Piecewise Potential (Quantum Dot)
1.2.4 The Nonlinear Schrodinger Equation
1.2.5 Heat Transfer in Vessel-Tissue System
1.3 Problems for Partial Differential Equations for Domains with Variable Boundaries
1.3.1 The Stefan Problem
1.3.2 Variable Pressure Calibrator (with Optimized Working Camera)
1.4 Inverse and Other Ill-Posed Tasks
1.4.1 Inverse Problem of Modeling Migratory Flows
1.4.2 Reconstructing Solutions for the Laplace Equation According to Measurement Data
1.4.3 Problem of Time Reversal for Heat Equation
1.4.4. Problem of Determining Boundary Condition
1.4.5. Problem of Temperature Field Continuation Based on Measurement Data
1.4.6. Air Pollution in Tunnel
Chapter 2: Selection of Functional Basis (Set of Bases)
2.1. Comparison of Functional Bases for Approximate Solution of Ordinary Differential Equations
2.2. Perceptron: Type of Tasks for which It Is Recommended to Apply with Specific Examples
2.3. RBF-Networks: Type of Tasks for which It Is Recommended to Apply with Specific Examples
2.4 Heterogeneous Networks Based on Perceptron and RBF
2.5. Multilayer Perceptron and RBF Networks with Time Delays
Chapter 3: Methods of Selection of Parameters and Structure for Neural Network Model
3.1. Structural Algorithms
3.2. Optimization Methods
3.3. Methods in Generalized Formulation
3.4. Methods for 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 for Domains with Fixed Boundaries
4.3. Solving Problems for Partial Differential Equations for Domains with Variable Boundaries
4.4. Solution of Inverse and Other Ill-Posed Problems
Chapter 5: Methods for Constructing Multilayer Semi-Empirical Models
5.1 General Description of Methods
5.1.1 Explicit Methods
5.1.2 Implicit Methods
5.2. Application of Methods for Ordinary Differential Equations
5.2.1. Stiff Differential Equation
5.2.2 The Mathieu Equation
5.2.3 Nonlinear Equation of Pendulum
5.2.4 Porous Catalyst
5.2.5 Sagging Hemp Rope
5.3 Application of Methods for Partial Differential Equations
5.3.1 Heat Equation
5.3.2. Problem of Deflection of Cloth under Weight
Appendices: Applications, Neural Network Technology, Thesaurus


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© Academic Press 2020
Academic Press
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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

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