Neural Networks: An Overview - Neural Networks: Methodology and Applications - page 16

Fig. 1.9. Interpolation of a parabola by a neural network with two hidden neurons;
( a )network;( b ) training set (+) and network output ( line ) after training; ( c ) out-
puts of the two hidden neurons (sigmoid functions) after training; ( d ) test set (+)
and network output ( line ) after training: as expected, the approximation is very
inaccurate outside the domain of variation of the inputs during training

k =1to N of a quantity of interest z p related to a physical, chemical,

financial, ... , process, are available. He assumes that there exists a relation
between the vector of variables

}

and the quantity z p , and he looks for a
mathematical form of that relation, which is valid in the region of variable
space where the measurements were performed, given that (1) the number
of available measurements is finite, and (2) the measurements are corrupted
by noise. Moreover, the variables that actually affect z p are not necessarily
measured. In other words, the engineer tries to build a model of the process
of interest, from the available measurements only: such a model is called a
black-box model. In neural network parlance, the observations from which the
model is designed are called examples . We will consider below the “black-box”
modeling of the hydraulic actuator of a robot arm: the set of variables { x }
has a single element (the angle of the oil valve), and the quantity of interest
{

{ x }

is the oil pressure in the actuator. We will also describe an example of
prediction of chemical properties of molecules: a relation between a molecular

z p }