Semi-empirical Neural Network Modeling and Digital Twins Development

Dmitriy Tarkhov, Alexander Nikolayevich Vasilyev

• Publisher: Academic Press
• Year: 2019
• Print ISBN: 9780128156513
• eBook ISBN: 9780128156520
• Topic: Biological Sciences
• Subtopic: Biotechnology
• Index: Biological Sciences

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

1

Examples of problem statements and functionals

Abstract

In this chapter, we begin a detailed presentation of our approach to the construction of neural network models on heterogeneous information, including differential equations, boundary, initial and other conditions, measurement data, etc. The first stage of our approach consists in the transition from the given information to the functional, which numerically characterizes the correspondence of the model to the available information about the simulated object. We described this stage in the first chapter, both in general form and by many specific examples. We have shown a large number of examples so that the reader can find an example that is most suitable for his task and make the necessary modifications to the functional we have considered. As examples, we considered both problems for ordinary differential equations and problems for partial differential equations, both simple problems and problems having complicating features. The problems with parameters, stiff, differential-algebraic, nonlinear, are considered as such features. The problems for composite domains with fixed and variable boundaries (both free and determined in the process of solving the problem based on the given criterion of optimality) are also studied. Also, we do not ignore ill-posed (in particular, inverse) problems in which measurement data replace all or part of the initial or boundary conditions. We describe the concrete results of computational experiments on the listed problems in Chapter 4.

Keywords

Ordinary differential equation;...

Table of contents

• Cover image
• Title page
• Table of Contents
• Copyright
• About the authors
• Preface
• Acknowledgments
• Introduction
• 1: Examples of problem statements and functionals
• 2: The choice of the functional basis (set of bases)
• 3: Methods for the selection of parameters and structure of the neural network model
• 4: Results of computational experiments
• 5: Methods for constructing multilayer semi-empirical models
• Index