## 5 Conclusions  
In conclusion, we propose a hybrid VMD-ESN method to predict the nonlinear and non-stationary sound pressure time series of aerodynamic noise induced by flow around a cylinder, addressing the challenges posed by complex turbulence-driven dynamics. The proposed method leverages the strengths of VMD and ESN. VMD adaptively decomposes the raw noise signal into quasi-orthogonal narrowband modes, effectively isolating frequency-specific turbulence features and mitigating non-stationarity. ESN processes each mode independently via a high-dimensional reservoir with sparse recurrent connections, where the echo state property enables dynamic memory of temporal evolution, and a linear output layer generates predictions several steps ahead. The predicted modes are finally aggregated to obtain the final prediction result, which enhances robustness against mode-specific uncertainties, achieving a 150-step prediction horizon with an RMSE of 1.05 Pa within the initial 50 steps. Validation using numerically simulated aerodynamic noise data demonstrates its capability to capture oscillatory instabilities critical for early warning in turbomachinery and aerospace systems. Future work could employ genetic algorithms to optimize network hyperparameters, yielding enhanced predictive performance, and wind tunnel experiments remain to be conducted to further validate the reliability of this methodology. The von Karman vortex mechanism and VMD's spectral adaptability establish physical-algorithmic coherence across bluff bodies, enhancing the model's predictive capability for diverse future applications. By bridging adaptive signal decomposition with temporal dynamics modeling, our work provides a practical tool for predicting spatiotemporal chaotic systems, with direct applications to noise control and disaster prevention in fluid-structure interaction scenarios.  
Methodology, Z.L. and H.M.; software, Z.L. and H.M.; validation, Z.L.; formal analysis, Z.L.; resources, J.Y., Z.L. and H.M.; data curation, J.Y. and Z.L.; writing--original draft preparation, J.Y., Z.L. and H.M.; writing--review and editing, J.Y. and Z.L.; supervision, J.Y., B.L. and J.C.; project administration, J.Y., B.L. and J.C.; funding acquisition, J.Y. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China (Grant Nos. 12174190), AI & AI for Science Project of Nanjing University, High-Performance Computing Center of Collaborative Innovation Center of Advanced Microstructures, and a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.  
Not applicable.  
The original data presented in the study are openly available at [[https://github.com/L-1818/vmd-esn](https://github.com/L-1818/vmd-esn)]([https://github.com/L-1818/vmd-esn](https://github.com/L-1818/vmd-esn)) (accessed on 10 June 2025).  
The authors declare no conflicts of interest.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Inflow Velocity** & **Time Steps Ahead** & **RMSE** & **R2** \\ \hline
60 m/s & 50 & 1.7600 & 0.9878 \\  & 150 & 3.9569 & 0.9544 \\
80 m/s & 50 & 1.2589 & 0.9854 \\  & 150 & 1.9671 & 0.9678 \\ \hline \end{tabular}
\end{table}
Table 2: Prediction performance of the VMD-ESN model.