## 2 C-POD Method and Modal Sorting Method
### Entropy-Controlled Euclidean-to-Probability Mapping Modal Sorting Method
### Evolutionary Characteristics of Modes Extracted by C-POD over Time

To further investigate the temporal evolution of modes extracted by C-POD, Figure 16 plots the variation of the first and second mode coefficients over time. The vorticity values in the cloud plot were restricted to the range \([-2,\,2]\) to facilitate comparison. It can be observed that the modal coefficients for both the first and second modes exhibited inflection points not only at the primary transition moments of the flow states (i.e., \(t=1\) s, 9 s, 33 s, and 110 s) but also at secondary fluctuation moments: \(t=5\) s, 13.5 s, 22 s, and 119.5 s.

Figure 16: Temporal variation of the first and second modal coefficients extracted by the C-POD method using the ECEPM method. Figure 15: Comparison of reconstruction accuracy for the first \(n\) modes extracted by POD and C-POD methods. This indicates that C-POD's time-varying modal coefficients effectively captured both dominant state transitions and subtle flow variations in the cylinder wake. The evolution of vorticity fields dominated by the first and second modes is illustrated at key time points: as shown in Figures 17-19, the time-resolved vorticity structures reflected the dynamic flow features governed by these leading modes. 1. For \(t<9\) s: * First mode: Dominated by third-type vortex, which weakened over time. * Second mode: Transitioned from irregular fourth-type vortex to first-type vortex, with rotation changes observed at \(t=1\) s, 5 s, and 9 s.
2. For \(9\) s \(<t<14.5\) s: * First mode: Evolved from third-type to first-type vortex, extending its influence across the field. * Second mode: Underwent gradual rotational transition. 3. For \(13.5\) s \(<t<33\) s: * The vorticity distribution became more stable. * Second-mode rotation direction changed. 4. For \(33\) s \(<t<110\) s: * System entered a quasi-stable regime, with negligible changes in vortex structures and modal vorticity values. 5. For \(110\) s \(<t<119.5\) s: * Flow transitioned toward vortex street formation. * Side vortices began shedding at the tail, and second-mode rotation altered again. 6. For \(t>119.5\) s: * The system reached a state of periodic vortex street generation. * Mode coefficients showed periodic oscillation (Figure 15). ## 5 Application Value of C-POD ROM in Inverse Problems

The Gappy POD method [31, 32] is a derivative of the POD reduced-order model, primarily used to solve the full-field reconstruction inverse problem when only sparse sensor data are available. The accuracy of this method heavily depends on the precision of the reduced-order model under low-order conditions. In both POD and C-POD methods, low-order modes represent the primary dynamics of the system, typically carrying most of the energy or important information, while high-order modes often represent finer details or

Figure 14.0 Temporal changes in the vorticity field dominated by the first and second modes of the C-POD method at \(t\) = 110, 119.5, and 135 s.

Figure 14.1 Temporal changes in the vorticity field dominated by the first and second modes of the C-POD method at \(t\) = 13.5, 22, and 33 s.

noise. For a system, low-order modes are typically associated with the main frequencies and significant physical behaviors, making them more accurate for data reconstruction. Through the analysis in Section 5, we found that, in the cylinder wake problem, the C-POD method had a greater advantage over POD at lower modal orders. This low-order advantage should be better reflected in inverse problems based on reduced-order methods. Continuing with the cylinder wake case described in Section 4.4.3, we describe the performance of this inverse problem in the cylinder wake scenario as follows: placing a specified number of sparse sensors in the vorticity field of the cylinder wake and reconstructing the vorticity field in real time based only on the sparse sensor data, to capture the complete vortex evolution of the cylinder wake.