Neural Networks: An Overview - Neural Networks: Methodology and Applications - page 65

of the process of glass forming. Since the phase diagrams of glasses exhibit
strong variations in the temperature domain of interest, many attempts at
such predictions have been made (see for instance [Kim 1991]), and databases
are available. Neural networks have been successfully used for the prediction
of liquidus temperatures [Dreyfus 2003], especially (as expected) for glasses
with more than three components.
Figure 1.42 illustrates, on the present industrial example, the parsimony of
neural networks. It shows scatter plots, which are very convenient for assessing
graphically the accuracy of the model: on the horizontal scale, the measured
value of the quantity of interest is displayed, whereas the predicted values are
displayed on the vertical scale. If prediction were perfect, all points should
be aligned on the first bisector; actually, due to measurement inaccuracy and
prediction errors, the points are more or less scattered; a good model should
generate equivalent scatterings for the points of the training set and those of
the validation or test set, and the vertical scattering should be on the order
of the standard deviation of the noise. Clearly, such a tool is no substitute to
the computation of the TMSE and VMSE as defined above, or of the leave-
one-out score defined in Chap. 2, but it provides a quick means of comparing
different models, for instance.
The model inputs are the contents of the glass in various oxides; the
output is the estimated liquidus temperature. Figure 1.42(a) shows the re-
sults obtained on a silica glass (made of potassium oxide K 2 O and aluminum
oxide Al 2 O 3 , in addition to silicon oxide SiO 2 , which is the main compo-
nent), obtained with a network having 6 hidden neurons (25 parameters), and
Fig. 1.42(b) shows the result obtained with a polynomial of degree 3, with
a similar number of parameters (19). Clearly, with roughly the same num-
ber of parameters, the neural network performs much better. For comparison,
Fig. 1.42(c) shows the scatter plot for a linear model.
1.4.8 An Application to the Modeling of an Industrial Process:
The Modeling of Spot Welding
Spot welding is the most widely used welding process in the car industry: mil-
lions of such welds are made every day. The process is shown schematically
on Fig. 1.13: two steel sheets are welded together by passing a very large cur-
rent (tens of kiloamperes) between two electrodes pressed against the metal
surfaces, typically for a hundred milliseconds. The heat thus produced melts
a roughly cylindrical region of the metal sheets. After cooling, the diameter
of the melted zone—typically 5 mm—characterizes the effectiveness of the
process; a weld spot whose diameter is smaller than 4 mm is considered me-
chanically unreliable; therefore, the spot diameter is a crucial element in the
safety of a vehicle. At present, no fast, nondestructive method exists for mea-
suring the spot diameter, so that there is no way of assessing the quality of
the weld immediately after welding. Therefore, a typical industrial strategy
consists