# Reduced-Order Model Based on Neural Network of Roll Bending
## 4 Neural Network
### Network 'Convexity'  
Table 1 presents several results of training the NN that predicts the value of the sheet convexity shown in Figure 8a on the basis of different inputs (rolling force \(F\), anti-bending force \(Q\), sheet width \(b\), and work and backup roll convexity \(c_{w}\) and \(c_{b}\)) for various variants of its structure. The network configuration shows the NN structure, the number of neurons in each layer beginning from the input layer (5 neurons), through one or more hidden layers, and to the output layer (1 neuron). The 'accuracy' column shows the percentage of variants in which the error is smaller than 0.005 mm.  
An analysis of training and testing for the latter option (NN7c) showed great agreement with the results of FEM modeling. Network testing was performed based on FEM calculations with the following values:  
* Rolling force \(F\): 6250, 18,750, 31,250 and 43,750 kN.
* Anti-bending force \(Q\): 1000 and 2000 kN.
* Combinations of convexity of work and backup rolls \(c_{w}\) and \(c_{b}\): 0.3-0.1; 0.25-0 and 0-0.25 mm.
* Sheet width \(b\): 1600 to 3400 mm every 200 mm.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Network**} & **Network Configuration** & **Mean Square Error** & **Correlation** & **Accuracy, \%** \\ \hline NN1c & 5-2-1 & 0.0082 & 0.9983 & 44.4 \\ \hline NN2c & 5-4-1 & 0.0079 & 0.9986 & 43.4 \\ \hline NN3c & 5-4-4-1 & 0.0049 & 0.9989 & 64.1 \\ \hline NN4c & 5-4-4-4-1 & 0.0057 & 0.9986 & 58.6 \\ \hline NN5c & 5-21-10-5-1 & 0.0061 & 0.9983 & 52.2 \\ \hline NN6c & 5-20-5-1 & 0.0059 & 0.9983 & 41.6 \\ \hline NN7c & 5-20-1 & 0.0029 & 0.9996 & 76.1 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Results of the training of the artificial neural network ‘convexity’ for various configurations.  
Figure 8: Variants of NNs for calculations of sheet convexity (**a**), rolling force (**b**), and anti-bending force (**c**).  
In total, 240 (=4 \(\times\) 4 \(\times\) 3 \(\times\) 10) results were obtained for the testing dataset. Figure 9 shows the results of the calculations obtained using the NN7c for the convexity of the sheet and an error for training and testing. The error in relation to the scope of changes in convexity in both cases is very small. The error is also shown in Figure 10.  
FNN is known to be very efficient for interpolation, but the accuracy of extrapolation is very low. Training of an FNN consists of reducing the error for the training dataset. A well-trained FNN gives proper results for the points between points of the dataset. An overtrained FNN gives better results for the training dataset, but worse results for the other points. To avoid overtraining, the testing dataset is used. When the error for the testing dataset begins to rise, it suggests the beginning of overtraining. Extrapolation is usually not provided for FNNs. The wider the range of the input and output values, the better the extrapolation properties, but the lower the accuracy or the more complex structure of the FNN. Then the range for inputs and outputs is usually 0.8-0.9 of maximum values. It means that the maximum value of, for example, the anti-bending force equal to 2500 kN should be 0.8-0.9 of the range of input values of the FNN. Values beyond the range are considered to have an equal value as on the boundary of the range. Therefore, no analysis beyond the range should be performed.  
The maximum error of the training dataset is 0.0451 mm; for the testing dataset it is 0.01066 mm. The standard deviation for the training dataset is 0.00629; for the testing  
Figure 10: Prediction error for training (**a**) and testing (**b**) datasets.  
Figure 9: Sheet convexity and prediction error for the training (**a**) and testing datasets (**b**).  
dataset it is 0.00296. The lower values of the testing dataset can be explained by the lack of extreme values of the input variable.