## 5 Conclusions  
In this paper, a 3D fuzzy online spatiotemporal multi-output support vector regression modeling method based on SGD is proposed for complex distributed parameter systems. The proposed method combines 3D fuzzy systems, support vector regression, and stochastic gradient descent methods and can adapt to system variations for online modeling. First, the proposed method effectively captures the complex nonlinear and spatiotemporal dynamical behavior of a DPS without reducing the model dimensionality, avoiding the loss of information caused by dimensional reduction. Second, a stochastic gradient descent algorithm was introduced to achieve efficient computation and the online update capability of the model. The effectiveness and applicability of the proposed method in complex nonlinear dynamic modeling were further validated through experimental verification using two typical distributed parameter systems and an actual industrial bottom furnace system. The experimental results show that the proposed method can accurately capture the spatiotemporal dynamic characteristics of the system, and exhibits good online modeling ability when facing real-time data and system changes. Compared with traditional modeling methods, the proposed method exhibits significant advantages in prediction accuracy, model update speed, and adaptability.  
Figure 28: TNAE comparison of the different methods in Case 3 (Reduction zone 1).  
Figure 29: RLNE comparison of the different methods in Case 3 (Reduction zone 1).  
However, the proposed method has some limitations. One potential limitation is that the performance of this method may decrease when dealing with extremely large datasets or high noise environments, as the SGD algorithm may have difficulty converging in such situations. In addition, although this method does not require dimensionality reduction, it may result in higher computational costs for high-dimensional systems. Future research may focus on improving the scalability of this method to more effectively handle larger datasets. Finally, applying this method to other types of industrial systems, such as real-time control systems, can further validate its universal applicability and effectiveness.  
Conceptualization, G.Z.; methodology, G.Z., X.Z. and H.Y.; software, G.Z. and H.Y.; validation, H.Y.; formal analysis, X.Z.; investigation, G.Z.; resources, H.Y.; data curation, G.Z.; writing--original draft preparation, G.Z.; writing--review and editing, X.Z.; visualization, H.Y.; supervision, B.W.; project administration, X.Z.; funding acquisition, X.Z. and B.W. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the project from the National Natural Science Foundation of China under Grant number 62073210 and Grant number 62173219.  
Not applicable.  
The original contributions presented in this study are included in the article. For further inquiries, please contact the corresponding author.  
The authors would like to thank the Editor-in-Chief, Editor, and anonymous Reviewers for their valuable reviews.  
The author [PERSON] was employed by the China Mobile Group Jiangsu Company Limited. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflicts of interest.