## 1 Introduction  
A distributed parameter system (DPS) refers to a dynamic system in which state variables are distributed simultaneously in time and space [1]. It is widely used in various industrial process applications, such as catalytic reactions [2], battery thermal processes [3], and chip curing [4]. The dynamic characteristics of these systems are described by partial differential equations (PDEs), showing significant spatiotemporal variation. The input, output, and process parameters change across both time and space dimensions [5]. Due to the nonlinear and spatiotemporal coupling characteristics of these systems, traditional lumped parameter system (LPS) simplification methods often cannot accurately capture their complex dynamic behaviors [6]. Traditional approaches often neglect the spatial distribution properties of a DPS and reduce it to an LPS, which leads to inaccurate modeling results. As the quality requirements of industrial products increase and sensor and computing technologies advance, the need for accurate modeling of distributed parameter systems has become increasingly urgent [7]. Advances in modern technology have enabled high-precision DPS modeling, which is essential to accurately describe the dynamical properties of the system. Accurate DPS modeling can not only improve the efficiency and reliability of industrial processes but also provide a solid basis for further research and applications [8]. Therefore, developing a mathematical model that can accurately describe the spatiotemporal dynamics of DPSs is an important task in current research. The modeling methods of distributed parameter systems are primarily categorized into two categories: process mechanism-based and data-driven [9]. Process mechanism-based modeling relies on physical and chemical principles to describe the system's dynamic behavior by establishing partial differential equations (PDEs). These PDEs are then simplified into finite-dimensional ordinary differential equations (ODEs) through model order reduction techniques, such as weighted residual methods [10], finite element methods [11], and finite difference methods [12]. These methods require that the PDEs are known. However, in practical industrial processes, it is often difficult to obtain an accurate PDE model due to the complexity of the system. In contrast, data-driven modeling-based approaches utilize large amounts of process data for system identification without prior knowledge, providing greater versatility and flexibility [13]. In particular, the data-driven distributed parameter system modeling approach methodology is more challenging to model real complex industrial processes when both the model structure and parameters are unknown [14]. In modeling approaches for distributed parameter systems based on spatiotemporal separation, common approaches assume spatial basis functions based on empirical or prior process knowledge [15]. The KL decomposition method is often used to construct spatial basis functions. [PERSON] et al. [16] proposed a multi-spatiotemporal modeling method that combines a finite Gaussian mixture model, KL decomposition, and principal component regression. This method effectively reduces nonlinear complexity by using spatial separation, local spatiotemporal modeling, and model integration. [PERSON] and [PERSON] [17] proposed a multimodal approach to successfully deal with the strong nonlinear and time-varying dynamics of the system by dividing the original runtime space through modified dissimilarity analysis, local spatiotemporal modeling using KL decomposition, and finally integrating the local model through principal component regression. Although the KL decomposition method has been successfully applied in many fields, its modeling ability for strongly nonlinear systems is limited due to its essentially linear nature. Therefore, KL decomposition is usually only applicable for modeling linear or weakly nonlinear systems. In addition, KL decomposition also has certain limitations in time-varying and nonlinear DPS modeling [18].