## 1 Introduction  
The spatial distribution characteristics are widely present in many manufacturing industries such as petroleum, chemical, steelmaking, and rolling. The dynamic characteristics of the system change with time and space, and this type of system is called a distributed parameter system (DPS) [1]. The dynamic characteristics of DPSs are usually described by partial differential equations and integral equations. Traditionally, the spatial distribution characteristics of such systems are often ignored, and the inherent DPSs are approximated as lumped parameter systems (LPS), and then are modeled and controlled using the mature lumped parameter methods. With the development in the times, people's requirements for product quality are gradually increasing. With the advancement of sensors, actuators, and computing technology, the modeling, control, and optimization techniques of DPSs have received increasing attention in the fields of science and engineering. For practical use in areas like the optimization, control design, analyzing, and numerical simulations of DPSs [2], it is necessary to use an appropriate mathematical model to describe its dynamic characteristics. Therefore, the modeling of DPSs has become a very important research field. The modeling methods for DPSs can usually be divided into mechanism modeling with known partial differential equations and grey-box/black-box modeling with unknown partial differential equations. Mechanism modeling is the process of obtaining partial differential equations for DPSs from the point of view of physical or chemical reaction knowledge of the system, such as thermal energy balance, material balance, or momentum balance relationships. Considering the complexity of calculations and the restricted quantity of sensors and actuators, modeling such systems further requires the use of spectral methods [1], finite elements [3],finite difference techniques, etc., to simplify the essentially infinite dimensional system into a finite dimensional system [4]. Nonetheless, in real-world scenarios, it will be challenging to obtain accurate mathematical descriptions of the system through the mechanism modeling method due to incomplete knowledge of physical and chemical processes. The second modeling method is based on data, including grey-box modeling and black-box modeling [5]. The model expressed by partial differential equations of the system is established through prior knowledge, and the unknown parameters of the model are obtained by identification from the input/output data of the system. This method is called grey-box modeling. Directly identifying the structure and parameters of a system from input/output data is called black-box modeling. In practical applications, black-box modeling has a wider range of applications. How to extract the relationship between system inputs/outputs with spatial distribution characteristics from data is currently a challenge in modeling research. In recent decades, researchers have conducted continuous research on black-box modeling of DPS. A few studies first discretized the PDE system in space and time, and then proposed the physical informed spark learning method to optimize the model parameters [6]. However, there is a linear relationship between the dimension of the system and the product of space-time discrete quantity, resulting in many model parameters. Most methods decompose variables into a series of spatial functions and temporal coefficients [7] and then reconstruct the system. Traditional methods usually preset spatial functions, such as Jacobian polynomials, spline functions, etc., and prior knowledge also dictates the quantity of spatial functions. The time coefficients are estimated using traditional modeling methods for LPS. In the past 20 years, spatial functions have been estimated using the Karhunen-Loeve (KL) decomposition method [8; 9]. The time coefficient is still estimated using traditional modeling methods for LPS.