## 2 C-POD Method and Modal Sorting Method
### Entropy-Controlled Euclidean-to-Probability Mapping Modal Sorting Method
While these two indicators, respectively, capture different aspects of reconstruction quality--trend similarity and numerical deviation--their combined use is not only less intuitive but also somewhat cumbersome in comparative analysis. Therefore, we introduced a new metric, Cr, defined in Equation (22), which integrates both correlation and error information to provide a more concise and comprehensive assessment of model performance:

\[\text{Cr}=\frac{\text{RMSE}}{\left|\mathbf{r}\right|} \tag{22}\]

where the root mean square error (RMSE) and the Pearson correlation coefficient \(r\) are defined as shown in Equations (23) and (24):

\[\mathbf{E}=\exp(\frac{\mathbf{Z}}{\tau})\]0

\[\mathbf{E}=\exp(\frac{\mathbf{Z}}{\tau})\]1

where \(n\) is the number of data points, \(y_{i}\) is the actual value, \(\hat{y}_{i}\) is the predicted value, \(\overline{y}\) is the mean of the actual data, and \(\overline{y}\) is the mean of the predicted data. A higher Cr value indicates an increasing trend in RMSE or a decreasing trend in the correlation coefficient, suggesting poorer reduced-order modeling performance. Under varying levels of modal truncation, the datasets were reduced using both the POD and C-POD methods. The resulting modal coefficients and orthogonal bases were then employed to reconstruct the original datasets. Figures 4 and 5 present the average and maximum values of Cr, computed using Equation (22). The average Cr value reflects the overall reduction capability, while the maximum value indicates the lower bound of this capability. As illustrated in Figure 4, both methods exhibited enhanced reduction performance with increasing modal truncation, stabilizing at higher modal orders. For modal orders below 6, the reduction capabilities of the two methods were comparable; however, for order values above this threshold, the C-POD method significantly outperformed POD. Figure 4 further shows that, regarding the lower bound of reduction capabilities, C-POD was superior at modal orders 7, 8, and 9, whereas POD maintained an advantage at other modal orders. These findings preliminarily highlight the superior reduction accuracy of the C-POD method. To compare the modal extraction capabilities of the C-POD and POD methods, we analyzed the reduced-order modeling of the Burgers' equation under the following conditions: \(c=0.1\), \(\beta=0\), and \(c=2\), \(\beta=1\). At times \(t=0\) and \(2\) s, both methods were applied to extract reduced-order modes, and the solutions reconstructed using only the first mode were analyzed. The temporal evolution of these first-mode-reconstructed solutions was tracked using the ECEPM method, as illustrated in Figures 6 and 7. \[\mathbf{E}=\exp(\frac{\mathbf{Z}}{\tau})\]2 Comparison of average Cr values under different modal orders: illustrating the overall reconstruction accuracy differences between POD and C-POD methods. \[\mathbf{E}=\exp(\frac{\mathbf{Z}}{\tau})\]3 Comparison of maximum Cr values under different modal orders: illustrating the minimum reconstruction accuracy differences between POD and C-POD methods. As observed from Figures 6 and 7, the first-mode-reconstructed solutions based on the C-POD method clearly exhibited time-dependent characteristics, including a gradual reduction in peak amplitude and forward propagation along the positive \(x\)-axis. In contrast, the first-mode reconstruction using the POD method showed minimal temporal variation, failing to capture the evolving features of the waveform. Further comparison revealed that, in the case of C-POD, the rate of peak amplitude decay was positively correlated with the parameter \(\beta\), while the propagation distance along the \(x\)-axis increased with wave speed \(c\). These results suggested that, compared to POD, the C-POD method provided more physically meaningful first-mode reconstructions that better reflected the temporal evolution of the system.