# Reduced-Order Model Based on Neural Network of Roll Bending
## 2 Research Structure  
Figure 1 presents a research flow chart related to the development, study, testing, and verification of the models presented in this paper. There are two kinds of model: the FEM model and ROM based on an NN.  
The roll bending FEM model and results of simulations are presented in detail in Section 3. The FEM model allows us to obtain distribution of elastic stresses and strains and displacement in the working and backup rolls. However, the shape of the roll surface in the exit plane is the more important information obtained in these simulations. The difference in vertical coordinates of two points located on the surface points in the exit plane defines the sheet convexity. These points are located in the center of the roll length and in the place that is in contact with the edge of the sheet. The results of FEM simulations with different  
Figure 1: Flowchart of research.  
parameters of the roll and rolling process (convexities of rolling and backup rolls, rolling and anti-bending forces, width of the sheet) are collected into datasets, which are used for the training, testing, and verification of the NN.  
There are many classifications of NN. Software-based NNs use a variety of topologies and learning algorithms: feedforward (FNN), convolutional, recurrent, time delay, modular, probabilistic, dynamic, group method of data handling, deep belief network, radial basis function network, memory networks, hybrid, and other types.  
The application of an NN in this paper can be referred to the problems of approximation or regression analysis. Another attribute of NN is the static problem that analyzes a state, not a dynamic process. In addition, NN should have supervised learning, which uses datasets that contain results from FEM simulations. These conditions eliminate most types of NN. For example, an adaptive NN presented in [32] and applied for trajectory tracking of robotic systems deals with process; representative points are self-dependent and require another type of NN than in this paper.  
In this paper, an FNN with supervised learning using least squares (backpropagation algorithm) is used. An FNN has a simple architecture and a relatively simple learning algorithm and is widely used for approximation or regression analysis for static problems. The neurons in an FNN are organized into layers: input, output, and hidden layers. The information in an FNN passes from layer to layer only in one direction (from the input through the hidden to the output layer) and does not pass inside the layer. In this paper, the simplest type of neuron (perceptron with a sigmoidal activation function) is applied to the FNN, allowing for the backpropagation learning algorithm. FNNs were chosen because of their usefulness and high efficiency in approximation problems.  
Three NNs are presented in Section 3 for the calculations of sheet convexity (NN'sheet convexity'), rolling force (NN 'rolling force'), and anti-bending force (NN 'anti-bending force'); that is, we will solve one direct and two inverse problems. Results of training, testing, and verification of these three NNs are presented. Unfortunately, it was not possible to perform experimental tests under laboratory or industrial conditions for validation. However, modeling using FEM is considered to be very accurate. Therefore, verification was limited by comparison with the FEM model. Furthermore, the comparison of NN 'convexity' and NN 'rolling force' can be treated as a cross-validation of the NN.