Page 182 - Neural Network Modeling and Identification of Dynamical Systems

5.2 SEMIEMPIRICAL ANN-BASED MODEL DESIGN PROCESS 173

FIGURE 5.2 Canonical form of the initial theoretical FIGURE 5.3 Canonical form of the initial theoretical
model (5.5) discretized by the explicit Euler method. model (5.6) discretized by the explicit Adams–Bashforth
method.
is the inclusion of additional time delay ele-
ments for the ODE right hand side values at pre- rows; thick arrows represent the connections
vious time instants. Canonical form of the recur- with varying weights.
rent neural network derived from the discretiza- Transition from a system of finite difference
tion of initial theoretical model (5.4)bythe Euler equations to a neural network model represen-
method (5.5) and the Adams–Bashforth method tation allows us to explicitly preserve the local-
(5.6) is shown in Figs. 5.2 and 5.3, respectively. ization of function, inherent to the original con-
The following notation is used in Figs. 5.2–5.9: tinuous time model as well as the correspond-
square-shaped elements represent external in- ing discrete time approximation. This feature
puts (control variables), measured outputs, and of semiempirical models allows us to “freeze”
time delay elements; round-shaped elements specific parts of the model, i.e., to prohibit the
represent the neurons; neuron activation func- modification of both its structure and the val-
tions are depicted schematically within the ues of its parameters during the training phase.
round-shaped elements (for instance, in this ex- This approach is justified for the models that
ample we use linear, quadratic, and sigmoidal include some relationships derived from accu-
functions); arrows between the elements repre- rate theoretical domain-specific knowledge, that
sent the corresponding input and output con- is, there is little doubt concerning these parts of
nections; arrows without the input element rep- the model. Other parts of the model that might
resent neuron biases; values of the biases and be doubtful (i.e., these parts are suspected to
connection weights are labeled along the ar- be the main reason of the low accuracy of the