## 4 Results and Discussion  
Prediction results of ESN, EMD-ESN, CEEMDAN-ESN, and VMD-ESN are compared in Figure 5. As shown in Figure 5a, it is observed that for the ESN model, deviation becomes clearly visible after about 25 time steps, and the model loses its ability to track the temporal evolution of the series over extended time steps. Both EMD-ESN and CEEMDAN-ESN broadly capture the oscillation patterns of aerodynamic noise time series within 300 time steps and track evolutionary trends more effectively than a single ESN. However, their prediction accuracy remains substantially limited. The results of the VMD-ESN model demonstrate that the prediction trend remains stable within the whole prediction time range, and good accuracy is achieved within 150 time steps. For the initial 50 time steps, as shown in Figure 5b, the absolute errors of the VMD-ESN model are consistently below 2 Pa, which are much smaller than those of other models. The error statistics of predictions by both models are recorded in Table 1. To comprehensively assess model accuracy, two distinct metrics are implemented: the root-mean-square-error (RMSE) and the determination coefficient (R\({}^{2}\)):  
\[\text{RMSE}=\sqrt{\frac{\sum\limits_{i=1}^{N}\left[y(i)-y_{\text{data}}(i) \right]^{2}}{N}}, \tag{10}\]  
\[\text{R}^{2}=1-\frac{\sum\limits_{i=1}^{N}\left[y(i)-y_{\text{data}}(i) \right]^{2}}{\sum\limits_{i=1}^{N}\left[y(i)-\overline{y}\right]^{2}}, \tag{11}\]  
where \(N\) is the number of predicted time steps, \(y(i)\) denotes the value at the \(i\)th time step in the predicted series, \(y_{\text{data}}(i)\) is the actual value at the \(i\)th time step, and \(\overline{y}\) is the average value of the time series. In Equation (9), as R\({}^{2}\) approaches 1, it indicates better fitting accuracy. For the ESN model, the RMSE reaches 5.15 Pa within the first 50 time steps, while the minus R\({}^{2}\) shows the incapability of its prediction for the series. VMD-ESN model achieves high accuracy with an RMSE of 1.0524 dB within 50 steps, which is substantially lower than the EMD-ESN model and CEEMDAN-ESN model. Its R\({}^{2}\) value of 0.9813 exceeds that of EMD-ESN, while CEEMDAN-ESN yields a negative R\({}^{2}\) indicating fundamental prediction failure. For 150-step prediction, the VMD-ESN model maintains robust performance with an R\({}^{2}\) of 0.9145, and the RMSE of it is 3.3225 dB, significantly outperforming the alternatives. This superiority stems from VMD's constrained variational optimization framework. Through bandwidth-constrained intrinsic mode extraction, VMD  
Figure 5: (**a**) Prediction results comparison of different models. (**b**) Absolute error distribution of the first 50 time steps. generates subseries optimally aligned with ESN's temporal processing. Consequently, VMD-ESN uniquely enables stable long-term prediction of high-Reynolds-number flow physics, making it suitable for aeroacoustic applications. It can be found that the proposed VMD-ESN method not only achieves a substantial enhancement in sound pressure time series prediction accuracy but also extends the reliable prediction horizon by approximately eight times compared to a single ESN model, effectively addressing the limitations of conventional ESN methods in long-term prediction. Compared with numerical simulation methods that require huge computing resources, the efficiency of our combined prediction model is greatly improved. The improvement in prediction accuracy and operational efficiency positions the VMD-ESN method as a practical solution for time series prediction of noise signals where both temporal resolution and computational resource constraints are critical considerations. To evaluate the generalization capability of the VMD-ESN model, two distinct cases with inflow velocities of 60 m/s (Re \(\approx\) 78,000) and 80 m/s (Re \(\approx\) 104,000) are tested.