Intelligent Systems II: Complete Approximation by Neural Network Operators

712 Pages·2016·6.54 MB·English

by  George A. Anastassiou (auth.)

Studies in Computational Intelligence 608
George A. Anastassiou
Intelligent Systems II:
Complete Approximation
by Neural Network
Operators
Studies in Computational Intelligence
Volume 608
Series editor
Janusz Kacprzyk, Polish Academy of Sciences, Warsaw, Poland
e-mail: [email protected]
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George A. Anastassiou
Intelligent Systems II:
Complete Approximation
by Neural Network Operators
123
George A.Anastassiou
Department ofMathematical Sciences
University of Memphis
Memphis,TN
USA
ISSN 1860-949X ISSN 1860-9503 (electronic)
Studies in Computational Intelligence
ISBN978-3-319-20504-5 ISBN978-3-319-20505-2 (eBook)
DOI 10.1007/978-3-319-20505-2
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To my wife Koula and my daughters
Angela and Peggy
Preface
This monograph is the continuation and completion of the author’s earlier 2011
monograph, “Intelligent Systems: Approximation by Artificial Neural Networks”,
Springer, Intelligent Systems Reference Library, Volume 19.
In this monograph we present the complete recent work of the last four years
oftheauthorinapproximationbyneuralnetworks.Itisthenaturaloutgrowthofhis
relatedpublications.Chaptersareself-containedandcanbereadindependentlyand
several advanced courses can be taught out of this book. An extensive list of
references is given per chapter.
ThetopicscoveredarefromAtoZofthisresearcharea,allstudiedforthefirst
time by the author. We list these topics:
Rate of convergence of basic neural network operators to the unit-univariate
case.
Rate of convergence of basic multivariate neural network operators to the unit.
Fractional neural network approximation.
Fractional approximation by Cardaliaguet-Euvrard and squashing neural net-
work operators.
Fractional Voronovskaya type asymptotic expansions for quasi-interpolation
neural network operators.
Voronovskaya type asymptotic expansions for multivariate quasi-interpolation
neural network operators.
Fractionalapproximationbynormalizedbellandsquashingtypeneuralnetwork
operators.
Fractional Voronovskaya type asymptotic expansions for bell and squashing
type neural network operators.
Multivariate Voronovskayatype asymptotic expansionsfor normalizedbell and
squashing type neural network operators.
Multivariatefuzzy-randomnormalizedneuralnetworkapproximationoperators.
Fuzzy fractional approximations by fuzzy normalized bell and squashing type
neural network operators.
Fuzzy fractional neural network approximation by fuzzy quasi-interpolation
operators.
vii
viii Preface
Higher order multivariate fuzzy approximation by basic neural network
operators.
High degree multivariate fuzzy approximation by quasi-interpolation neural
network operators.
Multivariate fuzzy-random quasi-interpolation neural network approximation
operators.
Approximation by Kantorovich and quadrature type quasi-interpolation neural
network operators.
Univariate error function-based neural network approximation.
Multivariate error function-based neural network approximations.
Voronovskaya type asymptotic expansions for error function-based
quasi-interpolation neural network operators.
Fuzzy fractional error function-based neural network approximation.
Highordermultivariatefuzzyapproximationbyneuralnetworkoperatorsbased
on the error function.
Multivariate fuzzy-random error function-based neural network approximation.
Approximation by perturbed neural network operators.
Approximation by multivariate perturbed neural network operators.
Voronovskaya type asymptotic expansions for perturbed neural network
operators.
Approximation by fuzzy perturbed neural network operators.
Multivariate fuzzy perturbed neural network operators approximation.
Multivariate fuzzy-random perturbed neural network approximation.
The book’s results are expected to find applications in many areas of applied
mathematics,computerscienceandengineering.Assuchthismonographissuitable
forresearchers,graduatestudents,andseminarsoftheabovesubjects,alsotobein
all science and engineering libraries.
The preparation of book took place during 2014–2015 in Memphis, Tennessee,
USA.
I would like to thank Prof. Alina Alb Lupas of University of Oradea, Romania,
for checking and reading the manuscript.
Memphis, USA George A. Anastassiou
March 2015
Contents
1 Rate of Convergence of Basic Neural Network Operators
to the Unit-Univariate Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Convergence with Rates of the Normalized Bell Type
Neural Network Operators. . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 The “Normalized Squashing Type Operators”
and Their Convergence to the Unit with Rates . . . . . . . . . . . . 8
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Rate of Convergence of Basic Multivariate Neural Network
Operators to the Unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Convergence with Rates of Multivariate Neural
Network Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 The Multivariate “Normalized Squashing Type Operators”
and Their Convergence to the Unit with Rates . . . . . . . . . . . 19
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3 Fractional Neural Network Operators Approximation . . . . . . . . 23
3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4 Fractional Approximation Using Cardaliaguet-Euvrard
and Squashing Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . 59
4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
ix
x Contents
4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.3.1 Fractional Convergence with Rates
of the Cardaliaguet-Euvrard Neural
Network Operators . . . . . . . . . . . . . . . . . . . . . . . . 64
4.3.2 The “Squashing Operators” and Their Fractional
Convergence to the Unit with Rates . . . . . . . . . . . . 77
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5 Fractional Voronovskaya Type Asymptotic Expansions
for Quasi-interpolation Neural Networks. . . . . . . . . . . . . . . . . . 89
5.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6 Voronovskaya Type Asymptotic Expansions for Multivariate
Quasi-interpolation Neural Networks. . . . . . . . . . . . . . . . . . . . . 103
6.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
7 Fractional Approximation by Normalized Bell and Squashing
Type Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
7.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
7.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
7.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
7.3.1 Fractional Convergence with Rates
of the Normalized Bell Type Neural
Network Operators . . . . . . . . . . . . . . . . . . . . . . . . 124
7.3.2 The “Normalized Squashing Type Operators”
and Their Fractional Convergence
to the Unit with Rates. . . . . . . . . . . . . . . . . . . . . . 134
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
8 Fractional Voronovskaya Type Asymptotic Expansions
for Bell and Squashing Type Neural Networks. . . . . . . . . . . . . . 143
8.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
8.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
9 Multivariate Voronovskaya Type Asymptotic Expansions
for Normalized Bell and Squashing Type Neural Networks. . . . . 153
9.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
9.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163