Engineering Optimization: Theory and Practice, Fourth Edition

13.7 Neural-Network-Based Optimization 729

Several neural network architectures, such as the Hopfield and Kohonen networks,
have been proposed to reflect the basic characteristics of a single neuron. These archi-
tectures differ one from the other in terms of the number of neurons in the network,
the nature of the threshold functions, the connectivities of the various neurons, and
the learning procedures. A typical architecture, known as themultilayer feedforward
network,is shown in Fig. 13.11. In this figure the arcs represent the unidirectional
feedforward communication links between the neurons. A weight or gain associated
with each of these connections controls the output passing through a connection. The
weight can be positive or negative, depending on the excitatory or inhibitory nature
of the particular neuron. The strengths of the various interconnections (weights) act as
repositories for knowledge representation contained in the network.
The network is trained by minimizing the mean-squared error between the actual
output of the output layer and the target output for all the input patterns. The error is
minimized by adjusting the weights associated with various interconnections. A number
of learning schemes, including a variation of the steepest descent method, have been
used in the literature. These schemes govern how the weights are to be varied to
minimize the error at the output nodes. For illustration, consider the network shown
in Fig. 13.12. This network is to be trained to map the angular displacement and
angular velocity relationships, transmission angle, and the mechanical advantage of a
four-bar function-generating mechanism (Fig. 13.9). The inputs to the five neurons in
the input layer include the three link lengths of the mechanism(r 2 , r 3 , andr 4 ) nd thea
angular displacement and velocities of the input link(θ 2 andω 2 ) The outputs of the six.
neurons in the output layer include the angular positions and velocities of the coupler
and the output links(θ 3 , ω 3 , θ 4 , andω 4 ) the transmission angle, (γ ), and the mechanical

Outputs

Inputs

Output

layer

Hidden

layer

Input

layer

Figure 13.11 Multilayer feedforward network. [13.23], reprinted with permission of Gordon
and Breach Science Publishers.