# Reduced-Order Model Based on Neural Network of Roll Bending
## 5 Discussion  
This paper presents an example of a four-high mill, though a similar analysis was performed for several four-high mills and simpler two-high mills. This methodology can also be applied for more complicated mills (six- to twelve- or twenty-high mills), but only in the 3D version of the FE model. Analytical methods of calculating the convexity of sheets, taking into account roll bending, have a number of simplifications that do not always sufficiently justify assumptions and cause one to receive only qualitative and inaccurate quantitative results. However, the finite element method, which gives high accuracy, cannot be used in real-time control systems because it requires long calculation times. However, the results of the FE modeling can be used to develop a simple model. The complex  
Figure 13: Dependencies of the anti-bending force, \(Q\), on the rolling force, \(\Delta F\) (**a**), dependence of the deviations of the anti-bending force, \(\Delta Q\), on the deviation of the rolling force, \(\Delta F\) (**b**).  
relationship between the convexity of the sheet and the many parameters and variables prevents the use of simple equations, thereby justifying the use of a more complex method, such as an artificial neural network.  
Unfortunately, it was not possible to perform experimental tests under laboratory or industrial conditions. The easiest way to carry out such a validation could be a measurement of the thickness distribution in the cross-section of sheets. Measurements should be fulfilled for several cases of the rolling and for several cross-sections of each sheet. Furthermore, such measurements could only be carried out at the beginning of the campaign until the shape of the rolls changed under the influence of temperature and wear. However, modeling using FEM is considered to be very accurate. Therefore, verification was limited by comparison with the FEM model.  
The 2D FE model gives enough good results (especially for two-high mills) and does not require much time for simulation. Of course, it is less accurate than a 3D model. The results of both the 2D and 3D models can be used to train a neural network. The inaccuracy of the FE model is directly transferred to the NN. Therefore, the appropriate strategy for the use of the FE models should be chosen.  
The fastest strategy is to use the 2D FE model and train the NN on these data. The entire training dataset presented in this paper was obtained in a time of less than 1 h. The model was run multiple times in the calculation, with several hierarchical cycles with various parameters on the PC. This was a sequential code for one-CPU core. The application of parallel computations on the CPU and more on the GPU accelerates the calculations multiplication, reducing the time to several minutes.  
The next strategy is 3D FE calculations. It is the most time-consuming approach. Sequential calculations cannot be used here, and parallel calculations should be organized. Calculations can be collected into some clusters. A cluster can contain calculations of the variants with the same stiffness matrix (the same convexity of work and backup rolls) and different loads (the rolling and anti-bending force, sheet width). It is the solution to the same equation but with a different right side. This strategy requires much more computational resources.  
A combined strategy. The 2D FE model serves for preceding or preliminary calculations that designate the domain of the permissible set of input parameters for further 3D simulations. For example, in this study, the dataset of 3575 points can be reduced to 2067 or less. This approach allows us to join together the previous two strategies. The 2D model rapidly gives a preliminary dataset for pre-training the NN, then the data will be replaced with more accurate results, and the NN will be re-trained.  
Future research will be directed toward the creation of a platform that can be used for the combined strategy and for communication between the FE model and the NN through the common dataset.