Page 175 - Neural Network Modeling and Identification of Dynamical Systems

166 5. SEMIEMPIRICAL NEURAL NETWORK MODELS OF CONTROLLED DYNAMICAL SYSTEMS

Another approach to semiempirical modeling • functions that belong to a general parametric
of dynamical systems is suggested in [7–15]. It family, everywhere dense in the space of con-
is also based on the combination of theoretical tinuous functions on compact subsets of Eu-
knowledge about the object of modeling and ex- clidean space (feedforward neural networks,
perimental data on its behavior. This interesting polynomials, etc.) which reflect the absence of
and promising approach allows us to deal with any a priori knowledge of some dependen-
objects that are described in the traditional ver- cies.
sion of both ODE and PDE cases.
Available experimental data is utilized as well
To describe our techniques, let us consider, at
first, the semiempirical approach to function ap- for the purpose of tuning the model parameters,
adjusting its structure to improve the accuracy
proximation. Recall that in the case of a purely and performing the adaptation in case the un-
empirical approach, the only assumption we known function is nonstationary.
make about the function to be approximated
In the general case, semiempirical models of
is its continuity. With this approach, we pick a
this kind need not be universal approximators
parametric family of functions which possesses
of continuous functions. However, they allow us
the universal approximation property (i.e., a set
to approximate functions of the specific type de-
of functions everywhere dense in the space of
fined by the model structure (which, in turn, is
continuous functions on compact subsets of Eu-
given by theoretical knowledge) up to any pre-
clidean space), such as the family of layered defined accuracy, as shown in the following the-
feedforward neural networks. Finally, we search orem.
this parametric family for the best approximant
according to some predefined objective function Theorem 1. Let m be a positive integer. Suppose
via numerical optimization methods. In the case that X i is a compact subset of R n x i , and Y i are com-
of a semiempirical approach, we assume that n y i for i = 1,...,m. Also suppose
pact subsets of R
some additional a priori knowledge is available that Z is a subset of R .Let F i be a subspace of
n z
for the unknown function apart from the con- the continuous vector-valued functions space from
tinuity. This knowledge is used to reduce the
ˆ
X i to Y i , and let F i be a set of vector-valued func-
search space, that is, to select a more specific
tions, everywhere dense in F i . Finally, let G be
parametric family of functions which simpli-
a subspace of Lipschitz continuous vector-valued
fies the optimization of an objective function.
ˆ
functions from Y 1 × ··· × Y m to Z, and let G be
In this way, we regularize the original problem
a set of vector-valued functions, everywhere dense
and reduce the number of free parameters for ˆ
the model, while preserving its accuracy at the in G. Then the set of vector-valued functions H =

ˆ
ˆ
ˆ
ˆ
same time. Thus, the semiempirical model is a ˆ g(f 1 (x 1 ),...,f m (x m )) | x i ∈ X i ,f i ∈ F i , ˆ g ∈ G ˆ is

parametric function family, whose elements are everywhere dense in the space H = g(f 1 (x 1 ),...,

the compositions of: f m (x m )) | x i ∈ X i ,f i ∈ F i ,g ∈ G .
• specific nonparametric functions which re-
Proof. Since the sets of vector-valued functions
flect the exact knowledge of some dependen- ˆ
cies; F i are everywhere dense in the respective spaces
• functions that belong to a parametric family F i for any vector-valued function f i ∈ F i and
any positive real ε i there exists a vector-valued
with specific properties (weighted linear com-
ˆ
ˆ
binations, trigonometric polynomials, etc.) function f i ∈ F i such that
which reflect the knowledge of general fea-
ˆ
tures of some dependencies; f i (x i ) − f i (x i ) <ε i , ∀x i ∈ X i .