## 1 Introduction  
Aerodynamic noise signals are strongly correlated with oscillatory instabilities in fluid dynamic systems that can trigger catastrophic events in applications [1, 2, 3]. Early detection of such instabilities through noise signal analysis is critical for preventing system-wide damage in areas of thermoacoustic instabilities, aeroacoustic resonance, and aeroelastic flutter. Existing methods for aerodynamic noise prediction predominantly rely on experimental measurements or numerical simulations [4, 5, 6, 7, 8, 9, 10], which are widely adopted in noise reduction optimization. Although these approaches can capture oscillatory instability mechanisms, they face significant limitations in practical implementation. Experimental methods demand strict wind tunnel environments to isolate aerodynamic noise, while numerical simulations require computationally intensive mesh generation and algorithm adaptation across diverse flow conditions. Such resource-intensive requirements hinder their efficiency and scalability, making them unsuitable for applications requiring real-time or long-term temporal behavior prediction of noise signals. With the rapid development of data science, machine learning methods have emerged as cost-effective alternatives to traditional high-fidelity analyses, offering more affordable approximations. These methods have gained significant traction in scientific domains, including fluid mechanics [11; 12; 13; 14; 15; 16; 17], nonlinear dynamics [18; 19; 20; 21; 22], and materials science and engineering [23; 24; 25; 26; 27; 28; 29]. [PERSON] et al. [16] proposed a data-driven deep neural network (DNN) method for fast prediction of aerodynamic noise with high accuracy. [PERSON] et al. [22] constructed recurrence networks to extract the peculiar features of phase space attractors in different fluid dynamic systems, which helped to detect the emergence of oscillatory instabilities. [PERSON] et al. [29] introduced a measurement technique based on recurrence networks for monitoring turbulent combustion transitions and predicting thermoacoustic instabilities and blowout events. Machine learning methods, proven effective in modeling complex systems, now enable reliable time series prediction by capturing nonlinear temporal dynamics in complex signals. To address the chaotic nature of underwater acoustic signals, [PERSON] et al. [30] proposed a model of gray wolf-optimized kernel extreme machine learning based on multivariate variational mode decomposition, achieving high accuracy in short-term predictions. [PERSON] et al. [31] proposed a model based on ensemble empirical mode decomposition-sample entropy (EEMD-SE) and full-parameters continued fraction for chaotic time series prediction, which enhanced both accuracy and practicality in wind power forecasting. However, accurate time series prediction of aerodynamic noise remains a persistent challenge due to its inherent broadband spectra and complex turbulence-driven dynamics, which complicate the modeling of temporal evolution under high-Reynolds-number flow conditions. Therefore, developing accurate prediction methods is critical for the early detection of oscillatory instabilities and the prevention of catastrophic events in fluid dynamic systems. Here, we propose a variational mode decomposition-echo state network (VMD-ESN) hybrid model for time series prediction of aerodynamic noise induced by flow around a cylinder to address the challenges attributable to the non-stationary features of the series, which aims to predict noise sound pressure with high efficiency and accuracy. The time series data of aerodynamic noise sound pressure is obtained through numerical simulations combining computational fluid dynamics (CFD) and computational aeroacoustics (CAA) methodologies. VMD adaptively decomposes the raw signal into multiple intrinsic mode functions (IMFs), which isolate multiscale turbulence features and reduce the redundancy of the whole system.