## 2 C-POD Method and Modal Sorting Method
### Entropy-Controlled Euclidean-to-Probability Mapping Modal Sorting Method
In the construction of the probability mapping function, the temperature parameter \(\tau>0\) is used to adjust the entropy of the probability distribution. The temperature parameter \(\tau\) plays a critical role in shaping the entropy level of the probability mapping. A smaller \(\tau\) (e.g., \(\tau\approx 0.1\)) leads to a more peaked distribution, favoring one dominant mode per snapshot, which increases mode separability but may result in instability. In contrast, a larger \(\tau\) (e.g., \(\tau>5\)) produces a flatter distribution, balancing contributions across multiple modes and improving robustness but potentially reducing interpretability. To balance these trade-offs, we recommend choosing \(\tau\) based on the desired sharpness of mode assignment. In this study, \(\tau\) was selected via grid search in the range [0.1, 5], and the optimal value was determined by evaluating the reconstruction performance using the Cr index. This approach ensures that the mode-ranking mechanism remains both physically meaningful and numerically stable. Through the above entropy-controlled Euclidean-to-probability mapping method, the probability weight matrix \(\mathbf{W}\) was obtained. For different samples in the database matrix, \(\mathbf{W}_{\mathrm{ij}}\) represents the probability value of the \(i\)-th clustering base for the \(j\)-th sample. A higher probability value indicates a greater contribution of that clustering base to the current sample value. Therefore, by examining the values in \(\mathbf{W}\), the importance of each clustering base in the C-POD method was distinguished. The construction process of the C-POD method and the entropy-controlled Euclidean-to-probability mapping method is shown in Figure 2. In Sections 3 and 4, to validate the effectiveness of the C-POD method, we applied it to two classic fluid dynamics problems: the one-dimensional Burgers' equation and the two-dimensional cylinder wake flow. These problems exhibited distinct nonlinear characteristics, making them ideal for testing the practical applicability of the method. The full procedure of the ECEPM is summarized in Algorithm 2. ``` Input: \(\mathbf{D}\in R^{m\times n}\)//Snapshot data matrix \(\mathbf{\mu}\in R^{m\times b}\)//Clustering bases (\(b\) modes) \(\tau>0\)//Temperature parameter for entropy control Output: \(\mathbf{W}\in R^{b\times n}\)//Probability weight matrix Steps: 1. Compute the squared norms of \(\mathbf{D}\) and \(\mu\): \(\text



## 3 Burgers' Equation and C-POD Method

### Introduction to the One-Dimensional Burgers' Equation

As a nonlinear partial differential equation, the Burgers' equation holds significant research value across various domains, including fluid mechanics [25] and meteorological forecasting [26]. In addition to characterizing the propagation of nonlinear waves, it encapsulates essential physical phenomena, such as viscosity and diffusion. Its inherent nonlinearity facilitates the simulation of complex flow phenomena, particularly in convection-dominated fields where highly nonlinear behaviors are prevalent. Dimensionality reduction methods aim to minimize computational costs while preserving the fundamental characteristics of the system. Consequently, using the Burgers' equation as a test case provides an effective means to evaluate the efficacy and accuracy of these models in addressing nonlinear and transient flow processes. In this study, the accuracy of the POD and C-POD methods was comparatively assessed using a one-dimensional Burgers' equation as the benchmark case. The mathematical formulation of the Burgers' equation is presented as follows:

\[\frac{\partial u(x,t)}{\partial t}+c\frac{\partial u(x,t)}{\partial x}=\beta \frac{\partial^{2}u(x,t)}{\partial x^{2}} \tag{19}\]

Let \(u(x,t)\) denote the wave amplitude, \(c\) represent the wave speed (i.e., the coefficient of the convection term), and \(\beta\) denote the dissipation coefficient.