## 2 C-POD Method and Modal Sorting Method
### Entropy-Controlled Euclidean-to-Probability Mapping Modal Sorting Method
Here, 80% of the original dataset was randomly selected as the training set, and the remaining 20% was used as the validation set. Using the correlation coefficient filtering method proposed by [PERSON] [33], seven optimal sparse sensor locations were selected in the vorticity field behind the cylinder. Specifically, this method computed the Pearson correlation coefficient between each candidate sensor point and the low-order POD modal coefficients obtained from the training set, then ranked and selected sensor locations with the highest absolute correlations. This ensured that the selected sensors were most sensitive to the dominant dynamic features of the flow field. These sparse sensors can output real-time vorticity information at their respective positions. Based on the sparse sensor data, the missing reconstruction methods based on POD (Gappy POD) and C-POD (Gappy C-POD) were used to reconstruct the cylinder wake flow field under different modal orders, and the reconstruction capabilities were measured, as shown in Figures 20 and 21. Figure 14.2 Variation in the average field reconstruction capability of Gappy POD and Gappy C-POD methods at different modal orders. From Figures 20 and 21, it is observed that the reconstruction accuracy of the Gappy POD method increased initially and then decreased as the modal order increased. When the modal order was 5, the Gappy POD method achieved its optimal reconstruction accuracy, with an average Cr value of 0.81 and a maximum Cr value of 0.97. When the modal order was less than 12, the reconstruction accuracy of the Gappy C-POD method followed a similar trend, increasing initially and then decreasing. When the modal order was 6, the average Cr value was 0.65 and the maximum Cr value was 0.84. However, when the modal order exceeded 12, the reconstruction accuracy of Gappy C-POD decreased as the modal order increased. When the modal order reached 20, the average Cr value dropped to 0.49, lower than the average value of 0.65 at modal order 6. This demonstrated the significant advantage of the Gappy C-POD method over the Gappy POD method in reconstructing inverse problems. The average reconstruction accuracy of Gappy C-POD was improved by 19.75% compared to Gappy POD, and the lower bound of reconstruction capability was improved by 13.4%. Furthermore, the Gappy C-POD method can achieve higher field reconstruction accuracy under higher modal order conditions, as higher modal orders can capture more complex flow behaviors. Therefore, the Gappy C-POD method is more suitable for field reconstruction in complex nonlinear systems. Based on the above analysis, a modal order of 5 was chosen for the Gappy POD method and a modal order of 6 for the Gappy C-POD method. The reconstruction of the cylinder wake vorticity field at \(t\) = 141 s is shown in Figures 22 and 23. From left to right, the three images in Figures 22 and 23 represent the cylinder wake vorticity field at \(t\) = 141 s, the reconstructed field with sparse sensor layout, and the distribution of reconstruction relative error. From Figures 21 and 22, it can be seen that the maximum error in the field reconstruction using the Gappy C-POD method was 25%, which is smaller than the maximum error of 35% in the field reconstruction using the Gappy POD method. Figure 14.3 Variation in the lower bound of field reconstruction capability for Gappy POD and Gappy C-POD methods at different modal orders. #### Limitations and Future Improvements

While the proposed C-POD ROM achieved improved reconstruction accuracy and enhanced interpretability compared to traditional POD-based methods, certain limitations remain, particularly in practical applications. * Computational Cost The integration of clustering and entropy-based ranking introduces additional computational burden. When dealing with large-scale datasets or increasing the number of clusters, both memory usage and computation time grow considerably.