## 2 C-POD Method and Modal Sorting Method
### Entropy-Controlled Euclidean-to-Probability Mapping Modal Sorting Method
* Sensitivity to Sensor Layout in Inverse Reconstruction For tasks involving sparse sensor reconstruction, the spatial placement of sensors significantly affects the performance. If the sensors fail to cover dynamically important regions, the model may suffer from decreased reconstruction fidelity. This highlights the necessity of coupling reduced-order modeling with sensor placement optimization. * Hyperparameter Dependency The performance of clustering and entropy-controlled ranking is sensitive to several hyperparameters, such as the number of clusters and entropy scaling coefficients. Inappropriate parameter settings can degrade the model's effectiveness or lead to unstable results. To address these limitations, future work may focus on: (1) Developing lightweight clustering algorithms or incremental learning schemes to reduce computational overhead. (2) Exploring adaptive sensor deployment strategies. (3) Investigating robust and data-driven hyperparameter tuning frameworks. These improvements will enhance the scalability and robustness of C-POD ROM in more complex and data-limited fluid dynamics scenarios. ## 6 Conclusions

This work introduced a clustering-based dimensionality reduction method guided by POD structures (C-POD), aiming to address the instability of clustering center initialization and the lack of effective mode ranking in conventional clustering-based approaches. By integrating the orthogonality and interpretability of POD with the flexibility of unsupervised

Figure 14.4 Gappy C-POD reconstruction of the cylinder wake vorticity field at \(t=141\) s.

Figure 14.5 Gappy POD reconstruction of the cylinder wake vorticity field at \(t=141\) s.

clustering, and by incorporating an entropy-controlled Euclidean-to-probability mapping (ECEPM) for robust mode evaluation, the proposed method enhanced both the stability and transparency of dimensionality reduction. The C-POD method was validated through applications to convective flow problems, including the one-dimensional Burgers' equation and the two-dimensional cylinder wake flow. The results demonstrated that C-POD effectively identified dominant flow structures and preserved essential dynamics under reduced-dimensional settings. Moreover, it showed strong resilience in sparse reconstruction tasks, delivering better performance when only limited sensor data or a small number of modes were available. Beyond improved numerical accuracy, the method provided meaningful insights into flow physics, making it well-suited for real-time prediction, control, and inverse analysis of complex unsteady systems. Future work will extend this framework to three-dimensional turbulent convective flows and explore adaptive clustering mechanisms that dynamically optimize cluster configurations, further enhancing model scalability and generalization. Conceptualization, Q.Y. and B.Z. ; methodology, Q.Y. ; validation, Q.Y. and B.Z. ; formal analysis, Q.Y. ; investigation, Q.Y. ; data curation, Q.Y. ; writing--original draft preparation, Q.Y. ; writing--review and editing, B.Z. ; visualization, Q.Y. ; supervision, B.Z. All authors have read and agreed to the published version of the manuscript. This research received no external funding. No new data were created or analyzed in this study. The authors declare no conflict of interest. ## References

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