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首页俄罗斯专家Galushkin的神经网络理论著作解析
"Neural Networks Theory" 是一本由Alexander I. Galushkin编写的关于神经网络理论的专业书籍,受到国际知名专家的高度评价。该书深入探讨了苏联和俄罗斯在神经网络研究领域的成果,对全球的研究者和实践者具有极高的价值。 这本书的特点在于它结合了数学方法、复杂性理论、非线性动力学和优化理论,这些都是俄罗斯传统科学的坚实基础。作者Galushkin教授不仅涵盖了神经网络架构等传统主题,还特别讨论了函数空间中的神经连续体,这在神经网络理论中是一个独特且广泛的领域。 书中包含176幅插图,旨在帮助读者更好地理解和掌握复杂的神经网络现象。此外,该书的出版得到了Springer出版社的支持,Springer是全球知名的科学出版机构,确保了内容的专业性和权威性。 值得注意的是,尽管本书的版权受到保护,但根据德国版权法的规定,经过许可后可以复制或部分复制该作品。违反这些规定可能会受到法律的追究。 专家如Robert Hecht Nielsen、Lotfi Zadeh和Sun-Ichi Amari都对该书给予了高度赞扬,认为它是神经网络理论的重要贡献,对理解神经网络的理论基础、发展和现状有极大的帮助。这表明该书不仅适合专业人士,也适合对神经网络感兴趣的学者和学生阅读,有助于拓宽对这一复杂领域知识的理解和应用。 "Neural Networks Theory"是一本深度探讨神经网络理论的著作,结合了多种理论框架和数学工具,为读者提供了一个全面且深入的视角来理解神经网络的工作原理和应用。无论是对于神经网络的初学者还是资深研究者,这本书都将是一个宝贵的资源。
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ContentsXVIII
11.4.5 Clusterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
11.4.6 Traveling Salesman Problem
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
11.4.7 Dynamic System Modelling
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
11.4.8 Conclusion
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
11.5 Typical Input Signal of Multilayer Neural Networks
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
Literature
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
12 Analysis of Closed-Loop Multilayer Neural Networks
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
12.1 Problem Statement for the Synthesis of the Multilayer Neural Networks
Adjusted in the Closed Cycle
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
12.2 Investigation of the Neuron Under the Multi-Modal Distribution
of the Input Signal
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
12.2.1 One-Dimensional Case – Search Adjustment Algorithm
. . . . . . . . . . . . . . . . . 224
12.2.2 Multidimensional Case – Analytical Adjustment Algorithm
. . . . . . . . . . . . . 226
12.3 Investigation of Dynamics for the Neural Networks of Particular Form
for the Non-Stationary Pattern Recognition
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
12.4 Dynamics of the Three-Layer Neural Network in the Learning Mode
. . . . . . . . . . . 235
12.5 Investigation of the Particular Neural Network
with Backward Connections
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
12.6 Dynamics of One-Layer Neural Networks in the Learning Mode
. . . . . . . . . . . . . . . . 242
12.6.1 Neural Network with the Search
of the Distribution Mode Centers f(x)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
12.6.2 Neural Network with N
*
Output Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
12.6.3 Neuron with K
p
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
12.7 Two-Layer Neural Network in the Self-Learning Mode
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
12.8 About Some Engineering Methods for the Selection of Matrix Parameters
in the Multilayer Neural Network Closed Cycle Adjustment Algorithms
. . . . . . . . 257
12.9 Design of the Multilayer Neural Network for the Matrix Inversion Problem
. . . 258
12.10 Design of the Multilayer Neural Network for the Number Transformation
from the Binary System into the Decimal One
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
12.11 Investigation of the Multilayer Neural Network
under the Arbitrary Teacher Qualification
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
12.12 Analytical Methods of Investigations of the Neural Network
Closed Cycle Adjustment
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
Literature
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
13 Synthesis of Multilayer Neural Networks with Flexible Structure
. . . . . . . . . . . . . . 273
13.1 Sequential Learning Algorithm for the First Neuron Layer
of the Multilayer Neural Network
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
13.2 Learning Algorithm for the First Neuron Layer of the Multilayer
Neural Network Using the Method of Random Search of Local
and Global Function Extrema
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
13.3 Analysis of Algorithm Convergence under the Hyperplane
Number Increase
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
13.4 Learning Algorithms for the Second Layer Neurons
of the Two-Layer Neural Network
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
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XIXContents
13.4.1 Condition of the Logical Function e(y) Realizability
Using One Neuron
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
13.4.2 Synthesis of a Neuron by the Functional Minimization Method
. . . . . . . . . 285
13.4.3 Neuron Synthesis by the Threshold Function Tables
. . . . . . . . . . . . . . . . . . . . . . 290
13.5 Learning Algorithm for Neurons of the Second and Third Layers
in the Three-Layer Neural Network
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
13.6 General Methods of the Multilayer Neural Network Successive Synthesis
. . . . . . . 292
13.7 Learning Method for the First-Layer Neurons of a Multilayer
Neural Network with a Feature Continuum
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
13.8 Application of the Adjustment Algorithm of the Multilayer Neural Networks
with Flexible Structure for the Problem of Initial Condition Selection . . . . . . . . . . 293
13.9 About the Self-Learning Algorithm for Multilayer Neural Networks
with Flexible Structure
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
Literature
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
14 Informative Feature Selection in Multilayer Neural Networks
. . . . . . . . . . . . . . . . . . 295
14.1 Statement of the Informative Feature Selection Problem
in the Learning Mode
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
14.2 About Structural Methods for the Informative Feature Selection
in the Multilayer Neural Networks with Fixed Structure
. . . . . . . . . . . . . . . . . . . . . . . . . . 297
14.3 Selection of the Initial Space Informative Features Using Multilayer Neural
Networks with Sequential Algorithms of the First-Layer Neuron Adjustment
. . 299
14.4 Neuron Number Minimization
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
14.5 About the Informative Feature Selection for Multilayer Neural Networks
in the Self-Learning Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
Literature
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
Part IV · Neural Network Reliability and Diagnostics
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
15 Neural Network Reliability
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
15.1 Methods for the Neural Network Functional Reliability Investigation
. . . . . . . . . . . 305
15.2 Investigation of Functional Reliability of Restoring Organs Implemented
in the Form of Multilayer Neural Networks
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
15.3 Investigation of Multilayer Neural Network’s Functional Reliability
. . . . . . . . . . . . . 308
15.4 Investigation of the Neural Network’s Parametrical Reliability
. . . . . . . . . . . . . . . . . . . 309
15.5 Investigation of the Multilayer Neural Network’s Functional Reliability
in the Case of Catastrophic Failures
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
Literature
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
16 Neural Network Diagnostics
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
16.1 Neural Network State Graph – The Main Notions and Definitions
. . . . . . . . . . . . . . . 322
16.2 Algorithm of Failure Localization in the Neural Networks
. . . . . . . . . . . . . . . . . . . . . . . . 323
16.3 Algorithm of the Minimum Test Design for the Failures
of the Logical Constant Type at the Neuron Outputs
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
16.4 Method of the Neural Network Adaptive Failure Diagnostics
. . . . . . . . . . . . . . . . . . . . 332
Literature
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
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ContentsXX
Part V · Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
17 Methods of Problem Solving in the Neural Network Logical Basis
. . . . . . . . . . . . 341
17.1 Neuromathematics – A New Perspective Part of Computational Mathematics
. 341
17.2 Neural Network Theory – A Logical Basis for the Development
of the Neural Network Problem Solution Algorithms
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
17.3 Selection of the Problems Adequate to the Neural Network Logical Basis
. . . . . . . 344
17.4 The General Structure of the Program Package for Problem Solution
in the Neural Network Logical Basis
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
17.5 Multilayer Neural Networks with Flexible Structure
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
17.6 Neural Network with Fixed Structure
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
17.6.1 Generation of the Input Signal of the Neural Network
. . . . . . . . . . . . . . . . . . . . 352
17.6.2 The Multilayer Neural Network Output Signal Generation
. . . . . . . . . . . . . . . 355
17.6.3 Formation of the Primary Optimization Criteria
. . . . . . . . . . . . . . . . . . . . . . . . . . 355
17.6.4 Selection of the Open Neural Network Structure
. . . . . . . . . . . . . . . . . . . . . . . . . . 356
17.6.5 Remarks about the Selection of the Open Neural Network Structure
that is Adequate to the Class of Solution Tasks
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
17.6.6 Remarks about the Activation Function Selection
. . . . . . . . . . . . . . . . . . . . . . . . . 358
17.6.7 Selection of the Multilayer Neural Network Structure
According to its Hardware Implementation Technology
. . . . . . . . . . . . . . . . . 359
17.6.8 Generation of the Secondary Optimization Functional
in the Multilayer Neural Networks
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
17.6.9 Generation of the Algorithm of the Search Procedure
for the Secondary Optimization Functional Extremum . . . . . . . . . . . . . . . . . . 360
17.6.10 Formation of the Adaptation Algorithms
in the Multilayer Neural Networks
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
17.7 Verification of the Adjusted Multilayer Neural Network
. . . . . . . . . . . . . . . . . . . . . . . . . . . 364
17.8 Elaboration of the Plan of Experiments
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
17.9 About the Importance of the Unification of Designations in the Process
of Synthesis of the Neural Network Adjustment Algorithms
. . . . . . . . . . . . . . . . . . . . . . 367
17.10 About Myths in Neural Network Theory
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
17.11 Conclusion
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
References
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376
Conclusion
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
Literature
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
Author’s Publications on Neural Network Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
Index
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
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I.1
Neural Computers
Neurocomputers are computers of a new class. Their appearance was determined for
two objective reasons: first, the principal stages of the development of modern elemental
base technology that mainly determines the development of computer architecture,
and second, practical requirements to solve specific problems in a faster and more
economical manner.
As far back as the 1950s, the main reason for neural computer development appeared
to be a development of the threshold logic that was permanently contradistinguished
to the classical development of the elemental base on the basis of AND, OR, NOT, etc.
This resulted in the implementation of a series of specific problem-oriented and ex-
perimental universal neural computers in the 1960s and 1970s. The terms “neural com-
puter” and “neurocomputer” are not associated with any feature or characteristic of
the human or animal nervous system. They are associated only with the conditional
name of a threshold element with the adjustable or fixed weights that implement an
elementary transfer function of the neural cell. A sharp upswing of LSI technological
development at the beginning of the 1970s, as well as the implementation primarily of
microprocessor chips with the classic computer architecture was realized on the el-
emental base on the basis of AND, OR, NOT, etc., resulting in the slowdown rather than
complete termination of the development of the computing facilities based on the
threshold logic elements. The next upswing at the beginning of the 1980s allowed one
to redefine the problem of neural computers due to the fact that VLSI technology, rather
than LSI technology, allowed one to implement in one or several chips not only a great
number of processing neuron elements, but also a whole set of connections between
them. This was not possible before. Such a possibility was provided by both electronic
as well as optical implementation methods in the middle of the 1980s.
The main idea of the neural computer construction, either a problem-oriented or
universal one, is to develop computers in the analog-digital form. In this regard, the
“fast” analog part performs multidimensional operations on the threshold basis. The
algorithms of the neural network coefficient adjustment are implemented either in the
“fast” manner in the analog form, or in the “low-speed” manner in the form of special-
ized digital circuits emulating neural algorithms, or in the “low-speed” manner in the
digital form, for example, using the universal personal computer.
The development of neural computers requires a design of the principally new
algorithms for the multidimensional solution of problems. The time for the solution
of the specific problem, on the one hand, only linearly depends upon the problem
Introduction
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2 I · Introduction
dimensionality and, on the other hand, is determined by the convergence time of the
iteration process for the solution in the particular neural network.
The main formal basis for the design of the neural algorithms is a neural network
theory [I-1 to I-7]. Let us call the main set of operations that are realized in the process
of the algorithm refinement as a “logical basis of the problem”. For the majority of
problems such a basis is the basis {Σa
_
x
_
}. The following problems require this basis: the
problems of vector algebra, Fourier transformation and optimization problems.
A class of tasks including solutions of the ordinary differential equations, Poisson,
Euler, Navier-Stokes equations, elliptical equations and so on can be reduced to the
aforementioned problems.
A logical basis of the computer system is the main group of operations implemented
by the elements of the basic operating device. In the case of classical computers, it is a
basis of AND, OR, NOT that forms, first, the level of more complex basis (Sheffer stroke,
multiple AND, OR, NOT, etc.), and a macro level, i.e., a level of microdevices. The logical
basis of the computer system is not determined by the logical basis of the solved problems
but requires an additional application of a rather complex program development system.
In the case of neural computers, the logical basis of the computer system in the
simplest case is the basis {Σa
_
x
_
, sign}. This basis maximally corresponds to the logical
basis of the major solved problems.
When solving problems with the neural computers’ logical basis, the basis of the
problem is in accordance with the basis of the computer system, and there are no ar-
tificial shifts in any direction:
one such shift corresponds to the problems with the threshold basis, whereas the
basis of the computer system is AND, OR, NOT;
another shift corresponds to the problem with the basis that differs from the thresh-
old one, whereas the computer system is neural.
It is supposed that the accordance between the computer system basis and the basis
of the problem provides the highest productivity. This statement is trivial in the case
of problem-oriented computers that are designed for the solution of a given specific
task. However, it is not trivial for the neural computers that pretend to be called uni-
versal at the present time.
All the difficulties of multiextremal search by iteration methods using neural com-
puters are evidently preserved. But these difficulties transfer from the software imple-
mentation (von-Neumann computers, computers with SIMD and MIMD architectures)
to the software/hardware implementation. In short, an algorithmic kernel of the main
array of applied problems can be realized in hardware or software/hardware form with
maximal operation speed. The neural computer is a maximally parallelized system for
a given algorithmic kernel implementation. The number of operation cycles in the
problem solving process, i.e., the number of the adjustment cycles for optimization of
the secondary functional in the neural computer, is not determined by the subjective
intuition of the circuit designer who distributes the processing among the circuit lay-
ers consisting of the Boolean elements. This number is also not determined by the
subjective views of the programmer who organizes the interaction between layers. On
the contrary, it is determined by the physical entity and complexity of the problem.
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