# Reduced-Order Model Based on Neural Network of Roll Bending
## 4 Neural Network
### Network 'Anti-Bending'  
The third NN is the NN 'anti-bending' (Figure 8c). The grounds for creating this network were the same as for the NN 'force'; that is, finding a solution to the inverse problem. The anti-bending force becomes the output, while the rolling force and the convexity are the inputs of the NN. Table 3 presents the NN 'anti-bending' results of training in various configurations. The 'accuracy' column shows the percentage of variants in which the error of calculating the anti-bending force is less than 500 kN (25%) for the first three networks (NN1a-NN3a), which were trained on the population of 3575 variants, and less than 100 kN (5%) for the remaining two networks (NN4a, NN5a). The results indicate the low accuracy of the first three networks. Then the number of variants was reduced to 2067, with a sheet convexity of not less than zero. An additional analysis shows the low influence of the anti-bending force on the convexity of narrow sheets. Therefore, variants with a sheet width smaller than 2500 mm were excluded, and the number of variants in the training dataset was reduced to 975. Figure 12 shows errors in the calculations of the anti-bending force for NN1a, NN4a, and NN5a. A network error with a smaller range of width and convexity compared to network NN1a is almost eight times smaller. The maximum error for network NN4a is equal to 253.5 kN in network 5--207.6 kN. The standard deviation for network NN4a is 59.8 kN, and for network NN5a it is 59.2 kN. NN 'anti-bending' was planned to be used in the real-time control system to control the convexity of the sheet when the inaccuracy of the rolling force should be compensated (disturbance) for by changes in the anti-bending force (output); that is, the NN 'anti-bending' \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Network**} & **Network** & **Mean Square** & \multirow{2}{*}{**Correlation**} & \multirow{2}{*}{**Accuracy, \%**} \\  & **Configuration** & **Error** & & \\ \hline NN1a & 5-10-1 & 0.1450 & 0.8125 & 67.4 \\ \hline NN2a & 5-20-1 & 0.1545 & 0.7824 & 63.3 \\ \hline NN3a & 5-20-5-1 & 0.1357 & 0.8394 & 70.9 \\ \hline NN4a & 5-10-1 & 0.01676 & 0.99776 & 94.8 \\ \hline NN5a & 5-20-5-1 & 0.01659 & 0.99781 & 94.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Results of training of the NN ’anti-bending’ for various configurations. Figure 12: Error of the NN ’anti-bending’: (**a**) network 1; (**b**) networks 4 and 5.  
should create a dependence of the anti-bending force on the rolling force for constants of other variables. The anti-bending force \(Q\) was limited to real values (from 0 to 2500 kN). The values of the input parameters were as follows: work and backup roll convexity \(c_{w}\) and \(c_{b}\): 0 and 0.3 mm. Sheet convexity \(c\): 0.1 and 0.2 mm. Sheet width \(b\): 2500, 3000, and 3500 mm. The rolling force, \(F\), was changed with a step of 500 kN in the range of 0 to 500,000 kN. Of the 24 possible variants, only the last condition was met. The plots for 14 variants are shown in Figure 12(a). An important conclusion that can be drawn from these plots is that all the plots have a similar shape and slope. This leads to the assumption that there is a dependence of the deviations of the anti-bending force, \(\Delta Q\), on the deviations of the rolling force, \(\Delta F\). To verify this assumption, an appropriate plot is shown in Figure 12(b). As a deviation of the anti-bending force, the difference between the real anti-bending and a half of maximum value of this force is calculated (maximum value of the anti-bending force is 2500 kN, and half of the maximum value is 1250 kN). The deviation of the rolling force refers to the difference between the actual rolling force and the base force, which corresponds to the anti-bending force of 1250 kN, with other factors held constant. Figure 12(b) proves the validity of the assumption.