## 2 C-POD Method and Modal Sorting Method
### Entropy-Controlled Euclidean-to-Probability Mapping Modal Sorting Method
The one-dimensional Burgers' equation incorporates both convection and diffusion characteristics, reflecting aspects of the heat conduction equation and the Navier-Stokes equations. Consequently, it establishes a conceptual link to both heat conduction and convection phenomena. Consider the initial condition in the form of a Gaussian pulse, as shown in Equation (20):

\[u(x,0)=\exp(-\frac{(x-x_{0})^{2}}{2\sigma^{2}}) \tag{20}\]

Under this initial condition, the analytical solution to the one-dimensional Burgers' equation is given by [27]:

\[u(x,t)=\frac{1}{\sqrt{1+\frac{2\beta t}{\sigma^{2}}}}\exp(-\frac{(x-x_{0}-ct) ^{2}}{2\sigma^{2}}) \tag{21}\]

Here, \(x_{0}\) denotes the initial peak position of the wave, \(\sigma\) represents the width of the wave packet, and c is the propagation speed (consistent with the meaning and value specified in Equation (19)). The analytical solution describes a Gaussian wave packet propagating in the \(x\)-direction with velocity \(c\). Notably, the width of the wave packet increases gradually over time, indicative of a diffusion phenomenon, while its amplitude decreases, reflecting a dissipation phenomenon that leads to gradual attenuation during propagation. A schematic illustration of the wave evolution in the one-dimensional Burgers' equation is presented in Figure 3. It is worth noting that although no artificial noise was explicitly added to the test data, numerical disturbances may naturally arise during simulation and data preprocessing. In light of this, the C-POD method framework incorporated proper orthogonal decomposition (POD) as the initialization step for clustering. By extracting dominant energy modes, POD inherently suppressed high-frequency perturbations and served as a natural denoising mechanism. Consequently, C-POD demonstrated a degree of robustness to noise throughout the training and reconstruction processes, enhancing its reliability in unsupervised reduced-order modeling. ### Database Construction

To evaluate the performance of the POD and C-POD methods, a dataset based on the Burgers' equation was first constructed. The initial conditions and parameters were set as follows:

Initial Conditions:

Initial wave peak position: \(x_{0}=0\)

Initial wave packet width: \(\sigma=1\)

Spatial and Temporal Discretization:

Spatial range: \(x\in[0,\,10]\), with \(Nx=1000\) uniformly distributed sampling points

Temporal range: \(t\in[0,\,5]\), with waveforms sampled at intervals of \(\Delta t=0.5\)

Independent Variable Settings:

Wave speed \(c\) range: \(c\in[0.1,\,2]\)

Dissipation coefficient \(\beta\) range: \(\beta\in[0,\,1]\)

For each pair of (\(c\), \(\beta\)), the Burgers' equation was solved at all time steps within the defined temporal range. At each time step, the spatial field \(u(x,t)\) was recorded. Each such spatial field--corresponding to a unique combination of (\(c\), \(\beta\), \(t\))--was treated as an independent snapshot. As a result, the snapshot matrix was constructed by stacking spatial fields obtained from different (\(c\), \(\beta\), \(t\)) combinations as column vectors. This resulted in a final snapshot matrix \(D_{1}\) of size \(1000\times 303\), where \(1000\) is the number of spatial points and \(303\) is the total number of snapshot instances across all parameters and time steps. This construction strictly followed the traditional POD framework, ensuring that each snapshot represented a single spatial field, with variations across snapshots arising from changes in either parameters or time--but not a mixture within a snapshot. Figure 3: Schematic of wave evolution in the one-dimensional Burgers’ equation. ### C-POD Method Accuracy Testing and Modal Decomposition

To evaluate the performance of the POD and C-POD methods, we conducted order reduction on the dataset and reconstructed the original data using various modal orders. The correlation coefficient and root mean square error (RMSE) were employed as evaluation metrics.