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CHAPTER
2

Dynamic Neural Networks: Structures

and Training Methods

2.1 ARTIFICIAL NEURAL To generate any model, we need to have at
NETWORK STRUCTURES our disposal:
• a basis, i.e., a set of elements from which mod-
2.1.1 Generative Approach to Artiﬁcial els are formed;
Neural Network Design • the rules used to form models by appropri-
ately combining the elements of the basis:
2.1.1.1 The Structure of the Generative • rules for the structuring of models;
Approach
• rules for parametric adjustment of gener-
The generative approach is widely used in ated models.
applied and computational mathematics. This 1
One of the generative approach variants is
approach, extended by the ideas of ANN mod-
that the desired dependence y(x) is represented
eling, is very promising as a ﬂexible tool for the as a linear combination of the basis functions
formation of dynamical system models. ϕ i (x), i = 1,...,n, i.e.,
The generative approach is interpreted fur-
n
ther as follows. We can treat the class of models
y(x) = ϕ 0 (x) + λ i ϕ i (x), λ i ∈ R. (2.1)
which contains the desired (generated) dynami-
i=1
cal system model as a collection of tools produc-
ing dynamical system models that satisfy some The set of functions {ϕ i (x)},i = 1,...,n, we will
speciﬁed requirements. There are two main re- call the functional basis (FB). The expression of
quirements for this set of tools. Firstly, it must the form (2.1) is a decomposition (expansion) of
generate a potentially rich class of models (i.e., it the function y(x) with respect to the functional
n
must provide extensive choice possibilities) and, basis {ϕ i (x)} i=1 .
We will further consider the generation of the
secondly, it should have as many as possible
simple “arrangement,” so that the implementa- FB expansion by varying the adjustable parame-
ters (the coefﬁcients λ i in the expansion (2.1)) as
tion of this class of models is not an “unbear-
able” problem. These two requirements, gener-
1 Examples of other variants are generative grammars from
ally speaking, are mutually exclusive. How and
the theory of formal grammars and languages [1–3], a syn-
by what tools to ensure an acceptable balance tactic approach to the description of patterns in the theory of
between them is discussed later in this section. pattern recognition [4–7].

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