# Reduced-Order Model Based on Neural Network of Roll Bending
## 1 Introduction  
The quality of the final product obtained in the rolling process depends on its preparation and realization. Both the preparation and the realization depend on the efficiency and accuracy of the models used. Real-time systems allow us to develop an effective rolling schedule, taking into account basic technological restrictions and optimization due to technical, economic, or other criteria: for example, the microstructure and mechanical properties of the final product. Real-time control systems, based on operative information about the condition of the process, should use not only accurate but also fast algorithms and models. More accurate modeling often takes into account more factors and less-important phenomena. The complexity and dimensions of the models are constantly arising, making them difficult or impossible to use in real-time processes. To reduce computational complexity, the order of the model can be reduced. This model replaces the original complex model and is commonly called a reduced-order model. The problem of roll bending under rolling loads in a four-high mill has been solved in two ways. The first was application of analytical solutions from the field of theoretical mechanics that were developed in the 1970s. The second was application of the FE method, starting with two-dimensional (2D) models and moving with the development of computer technology to three-dimensional (3D) models. At the turn of the century, the calculation time for one roll bending case using a 2D model was 10-15 s, and the 3D model on a supercomputer was 10-15 min, with an effectiveness of cases of 70-80%. Now, the calculation time on a personal computer (PC) for one 3D case is 1 to 2 min, and for the 2D FEM, it is 1 s. However, for real-time control systems, this time should be one or two orders of magnitude less. Model order reduction is achieved by approximation, reducing the state or spatial dimension, or decreasing the degrees of freedom of the original model. Taking into account the roll bending problem and the presence of very accurate modeling tools, which are FEM, it was decided to apply approximation for the model order reduction. The main requirements for ROM are computational efficiency, a small approximation error, and conservation of the properties and characteristics of the original model [1]. Model order reduction methodologies are classified into such classes:  
* Data-driven modeling [2; 3; 4; 5; 6], including machine learning and deep learning, part of which are neural networks;
* Proper orthogonal decomposition (POD) used for computational fluid dynamics and structural analysis [7; 8];
* Proper generalized decomposition is used to solve boundary value problems [9; 10];
* Reduced basis methods (RBM) applied for nonlinear, steady-state thermal analysis [11], finite volume approximations [12], the Navier-Stokes equation [13], stochastic problems [14], etc. ;
* Simplified physics [15; 16];
* Matrix interpolation [17; 18; 19];
* Interpolation with the transfer function [20] and the other. These methodologies have been analyzed from the point of view of suitability to the problem being considered. Data-driven modeling seems very feasible for this problem, and analysis of an NN is presented in the next section. POD is a powerful and effective approach to data analysis aimed at identifying low-order modes of a system. The key goal of POD is to reduce a large number of interdependent variables to a much smaller number of uncorrelated variables. The POD is applied for dynamic systems, and its research areas are as follows: fluid mechanics, signal processing, image processing, rotor dynamics, thermodynamics, structure dynamics, microelectromechanical system, chemical reactions, control, optimal control, electrical engineering, etc. POD deals with processes, mainly with the use of partial differential equations. In contrast, the problem considered in the paper is static and deals with the state, not the process. That is why POD has now been applied to model roll bending.