## 4 Case Studies
### Case 1: Nonisothermal Fixed-Bed Reactor  
In the actual industrial production process, the measured data are subject to noise. To simulate this, Gaussian white noise was added to the output of the system model to align with real-world conditions. The noise was modeled as random noise with a uniform distribution within the range of \([-0.02,0.02]\), ensuring a mean value of 0 and a bounded amplitude. The noise signal was generated using a random number generator, with each sample independently drawn at each time step to simulate the typical measurement noise encountered in industrial systems. The excitation time was set to 40 s, with a sampling interval of 0.2 s, resulting in a total of 200 data samples collected for the initialization of the model. During the online simulation, the total modeling time was set to 50 s, generating 250 data samples. After 10 s of online operation, to simulate environmental changes in the system, the reaction rate was increased by reducing the activation energy. Specifically, the activation energy was reduced to 10% of its original value after 10 s of operation, mimicking the impact of changing environmental conditions on the process. In this section, a nonisothermal fixed-bed reactor system is modeled, as shown in Figure 3. The gaseous reactants are injected into the reactor, and the reaction from A to B occurs on the solid catalyst. The reaction is endothermic, so a jacket is required to heat it. Assuming that the environment is under ideal conditions, the mathematical model of the dynamics of this reaction process can be formulated as follows:  
\[\left\{ \begin{matrix}\epsilon_{p}\frac{\partial T_{g}}{\partial t}=\frac{ \partial T_{g}}{\partial z}+\alpha_{c}\big{(}T_{s}-T_{g}\big{)}-\alpha_{g} \big{(}T_{g}-u\big{)}\\ \frac{\partial T_{s}}{\partial t}=\frac{\partial^{2}T_{s}}{\partial z^{2}}+B_ {0}\text{exp}\Big{(}\frac{\gamma T_{s}}{1+T_{s}}\Big{)}-\beta_{c}\big{(}T_{s} -T_{g}\big{)}-\beta_{P}(T_{s}-b(z)u)\end{matrix} \right. \tag{30}\]  
The boundary conditions and the values of the process parameters are as follows:  
\[\left\{ \begin{matrix}z=0,T_{g}=0,\frac{\partial T_{g}}{\partial z}=0\\ z=1,\frac{\partial T_{s}}{\partial z}=1\end{matrix} \right. \tag{31}\]  
\[\epsilon_{p}=0.01,\gamma=21.14,\beta_{c}=1.0,\beta_{P}=15.62,\beta_{0}=-0.003,\alpha_{c}=\alpha_{g}=0.5 \tag{32}\]  
where \(t\) and \(z\) are dimensionless time and space. \(T_{g}\), \(T_{s}\), and \(u\) are dimensionless gas temperature, catalyst temperature, and jacket temperature, respectively. Figure 3: Nonisothermal fixed-bed reactor. An interference signal with an amplitude of 10% is added to the manipulated variables \(u_{1}\), \(u_{2}\), and \(u_{3}\) to collect spatiotemporal data with sufficient information from the system as follows:  
\[ \begin{split} u_{1}(t)&=0.7114+0.1*0.7114*rand()*sign( rand()-0.5)\\ u_{2}(t)&=0.6858+0.1*0.6858*rand()*sign(rand()-0.5)\\ u_{3}(t)&=0.6838+0.1*0.6838*rand()*sign(rand()-0.5) \end{split}  \tag{33}\]  
where 0.7111, 0.6858, and 0.6838 are the steady-state inputs of the catalyst temperature under the spatial reference contour \(T_{\text{sd}}(z)=0.42-0.2\text{cos}(\pi z)(0\leq z\leq 1)\). This reactor can only measure the catalyst temperature, which is measured using a \(p\)-point sensor. In this example, seven sensors were used and uniformly placed. The dynamic mathematical model described by (30) and (31) was used to reveal the thermal dynamic characteristics of the system and was used as a simulation model to verify the performance of the modeling algorithm. The spatially continuous temperature estimated output of the nonisothermal catalytic packed-bed c reactor system is shown in Figure 3(a), which is in high agreement with the measured temperature output in Figure 3(b). In Figure 5, the predicted temperatures at the sensors \(S_{3}\) and \(S_{7}\) are compared with the actual temperatures. It can be seen that the proposed method performs well. Figure 4: Prediction results of 3D-OMSVR-SGD for nonisothermal catalytic packed-bed reactors.