# Reduced-Order Model Based on Neural Network of Roll Bending
## 1 Introduction
Proper generalized decomposition is used to solve boundary value problems, often described by partial differential equations. Roll bending is not a boundary value problem. RBM is a technique used to create reduced-order models of parametrized partial differential equations. This means that it also deals with processes, not states, and it cannot be applied to model roll bending. Methods of simplified physics use linearization, approximation, breaking down complex concepts into smaller, digestible parts, and reducing state spatial dimensions. The methods leave the physical foundations unchanged. Reduction of the 3D problem to 2D is an example of such a method presented in the paper. Common techniques of matrix interpolation are linear, polynomial, spline interpolation, and the specialized method. The goal of this is to estimate the value of a matrix at a point between known matrix values, often based on a set of discrete data points. The main applications are signal processing, computer graphics, data analysis, control systems, and FEM. FEM is based on such matrix interpolation that defines material properties, displacements, temperature, or stress-strain fields, and so on. The approximation with neural networks can be treated as a specialized method, but it is not. Interpolation with the transfer function treats process as an object and describes it with transfer functions (differential equations after the Laplace or Laplace-Carson transformation). The problem considered in the paper is static. In summation, it can be noted that the most useful methods are data-driven, with NN approximation. The rolling process is one of the processes that requires fast and accurate models for efficient real-time control. The improved mechanical properties of many advanced steel grades are achieved on the basis of the concept of thermomechanical treatment or thermomechanical process control. Recrystallization, precipitation, and phase transformation phenomena are modeled and controlled to produce a fine homogeneous microstructure that results in improved properties. Computer-assisted, real-time schedule design, modeling, and control require fast and accurate ROM [21]. The rolling process is multiscale, multivariable, nonlinear, and unbalanced, with strong coupling and a non-steady state [22]. High-speed rolling conditions and rapid changes make controlling it difficult. To obtain accurate control of large-scale and high-speed rolling, a multi-objective, real-time calculation method for rolling schedules should be introduced. Rolling process analysis based on large-scale industrial data should be taken into account to establish dynamic process models. [PERSON] et al. [23] developed a fast rigid plastic FE method for online application in strip rolling and showed that this method met both computational time and accuracy requirements. There, the energy method was proposed to predict the rolling force with a refined neural network. [PERSON] et al. [24] stated that the first description of neural network applications in steel rolling was published in 1991 and since then the number of publications has steadily increased. The backpropagation algorithm for NN training is the main one. NNs allow us to investigate whether new products of a certain width, strength, and thickness can be produced and to determine optimal mill settings. Other NN application models and data processing can predict the mechanical properties of the steel strip and the rolling force. It can be used to determine the pre-setting of the mill. Another real-time model can accurately predict the rolling force and rolling power in the strip rolling mill [25]. The model uses multiple studies of the process with FE simulations and the choice of parameters that describe the thermomechanical behavior of the strip. [PERSON] et al. [26] and [PERSON] and [PERSON] [27] proposed a long-term learning method using an NN to improve the accuracy of rolling force prediction in a hot rolling mill.