## 2 Problem Description  
### Mathematical Description of Distributed Parameter Systems and Its Characteristics  
Numerous processes, like the industrial chemical reaction [1], semiconductor production [17], transmission in cure or reflow processes [18] and Lithium-ion Battery Thermal Process [19; 20], are spatially distributed in practice. They can be depicted using the subsequent mathematical formula:\[\begin{array}{l}\frac{\partial^{m}w(f,t)}{\partial t^{m}}=\gamma w(f,t)+h(w)+ \varpi b(f)Q(t),\\ l_{a}\leq f\leq l_{b},t\geq 0,\\ m=1\text{ or }2.\end{array} \tag{1}\]  
where \(w(f,t)\) represents the output; \(h(w)\) is a nonlinear function; Y is the linear space differential operator, which includes first- or second-order spatial derivatives (\(\partial/\partial f,\partial^{2}/\partial f^{2}\)), and is dense in Hilbert space; \(\varpi\) is a constant vector; \(Q=\left[q_{1}\ q_{2}\ \cdots\ q_{\mathbb{L}}\right]\in R^{1\times L}\) is manipulating the input vector (i.e., multiple control sources); and \(b(f)=\left[b_{1}(f)\ b_{2}(f)\ \cdots\ b_{\mathbb{L}}(f)\right]^{T}\in R^{ \mathbb{L}}\) is a known smooth function about \(f\), which describes the distribution of \(Q\) in the spatial domain \(\bar{F}\subset\left[l_{a},l_{b}\right]\).  
Usually, boundary and initial conditions also restrict the system depicted in Equation (1). As an illustration, at \(t=1\), the boundary and initial conditions can be depicted in the following equations:  
\[\begin{array}{l}\frac{\partial w(l_{a},t)}{\partial f}=g_{a}(w(l_{a},t),Q(t )),\\ \frac{\partial w(l_{b},t)}{\partial f}=g_{b}(w(l_{b},t),Q(t)).\end{array} \tag{2}\]  
\[w(f,0)=w_{0}(f). \tag{3}\]  
where \(g_{a}(w(l_{a},t);Q(t))\) is dependent on \(w(l_{a},t)\) and \(Q(t)\); \(g_{b}(w(l_{b},t)Q(t))\) is dependent on \(w(l_{b},t)\) and \(Q(t)\); and \(w_{0}(f)\) is a known function.  
This type of system has spatiotemporal coupling characteristics and is an infinite dimensional system. \(w(f,t)\) can be spatiotemporally separated into an infinite weighted sum of space base functions and time coefficients as shown in Equation (4):  
\[w(f,t)=\sum_{i=1}^{\infty}w_{i}\varphi_{i}(f). \tag{4}\]  
where \(\varphi_{i}(f)\) is a space base function and \(w_{i}\) is a time coefficient.  
Practically, it is necessary to utilize a finite quantity of sensors positioned at \(f_{1},f_{2},\cdots,f_{P}\); therefore, \(\bar{F}=\left[f_{1},f_{2},\cdots,f_{P}\right]\). \(w(\bar{F},t)=\left[w(f_{1},t)\,w(f_{2},t),\cdots,w(f_{p},t)\right]\) represents the output in space, where \(P\) is the number of sensors. The data-driven modeling challenge for the DPS as outlined in Equations (1)-(3), entails the task of determining a spatiotemporal coupled model using the input \(\{Q(k)\}_{k=1}^{K}\) and the output \(\{w(f_{i},k)\}_{i=1,k=1}^{PK}\), where \(K\) is the time length. Essentially, systems with an infinite number of dimensions will be converted into systems with a finite number of dimensions. When the mechanism model described in Equations (1)-(3) is unknown, utilizing the KL decomposition technique, the system's space base function can be derived from the gathered data set \(\bar{S}=\{Q(k)\ w(f_{i},k)\}_{i=1,k=1}^{P,K}\). Then, \(w(f,t)\) can be approximately depicted as Equation (5):  
\[w_{n}(f,t)=\sum_{i=1}^{n}w_{i}\varphi_{i}(f). \tag{5}\]  
The time coefficients in Equation (5) can be identified using traditional methods.  
However, the modeling approach of DPS using KL decomposition unavoidably results in neglected dynamic features and uncertainties in the identified model due to the decrease in dimensions.