Time Series Prediction of Aerodynamic Noise Based on Variational Mode Decomposition and Echo State Network

Abstract

Time series prediction of aerodynamic noise is critical for oscillatory instabilities analyses in fluid systems. Due to the significant dynamical and non-stationary characteristics of aerodynamic noise, it is challenging to precisely predict its temporal behavior. Here, we propose a method combining variational mode decomposition (VMD) and echo state network (ESN) to accurately predict the time series of aerodynamic noise induced by flow around a cylinder. VMD adaptively decomposes the noise signal into multiple modes through a constrained variational optimization framework, effectively separating distinct frequency-scale features between vortex shedding and turbulent fluctuations. ESN then employs a randomly initialized reservoir to map each mode into a high-dimensional dynamical system, and learns their temporal evolution by leveraging the reservoir’s memory of past states to predict their future values. Aerodynamic noise data from cylinder flow at a Reynolds number of 90,000 is generated by numerical simulation and used for model validation. With a rolling prediction strategy, this VMD-ESN method achieves accurate prediction within 150 time steps with a root-mean-square-error of only 3.32 Pa, substantially reducing computational costs compared to conventional approaches. This work enables effective aerodynamic noise prediction and is valuable in fluid dynamics, aeroacoustics, and related areas.

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]. Meng et al. [16] proposed a data-driven deep neural network (DNN) method for fast prediction of aerodynamic noise with high accuracy. Ruiz 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. Godavarthi 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, Yang 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. Wang 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. Each mode is confined to a narrowband frequency spectrum, ensuring the decomposition aligns with the intrinsic dynamic features of the signal. Subsequently, ESN processes each mode independently by mapping its temporal dynamics into a high-dimensional state space. The echo state property of the reservoir enables it to retain memory of past inputs, allowing dynamic modeling of evolution for each mode. All the predicted individual modes are combined to generate the final result. The proposed VMD-ESN method achieves good fitness within 150 time steps and demonstrates a substantial reduction in computational resource requirements compared to conventional numerical approaches.

2. VMD-ESN Model

The framework of the proposed VMD-ESN model is shown in Figure 1. In the model, VMD is employed as the initial step to decompose the noise signal into a series of intrinsic mode functions. While a noise signal exhibits non-stationary behavior, its statistical features change over time. VMD handles the problem by decomposing it into modes that vary in frequency and amplitude over time by establishing a variational optimization framework that iteratively refines the decomposition to best fit the time-varying characteristics of the signal. Meanwhile, with the reconstruction constraint, VMD minimizes the sum of the squared frequency bandwidths of all modes to make the bandwidth of each mode as narrow as possible, ensuring that the modes reflect the essential features of the original noise signal, which has complex frequency structures brought by vortex-induced vibrations and turbulence-associated fluctuations. Then, each mode generated by VMD is fed into ESN as an input vector. As a special type of recurrent neural network, ESN uses a large reservoir with numerous neurons, enabling it to save information of previous inputs, which means it has a great short-term memory. Also, sparse connectivity within the reservoir facilitates its decomposition into multiple loosely coupled subsystems, enriching its echo signals and dynamic structure. With its special structure, ESN is used to capture features of all modes and predict their future evolution. Finally, the predictions from all the individual modes are combined to generate the predicted output for the original time series.

2.1. Variational Mode Decomposition

The VMD process seeks to minimize the total bandwidth of all modes while maintaining perfect signal reconstruction. Essentially, VMD formulates the signal decomposition as a constrained variational optimization problem, which can be mathematically expressed as follows [32]:

$$\left\{\begin{array}{l}\underset{\left\{u_k\right\},\left\{{\omega }_k\right\}}{\min }\left\{{\displaystyle \sum _k{‖{\partial }_t\left[\left(\delta \left(t\right)+\frac{j}{\pi t}\right)u_k\left(t\right)\right]{\mathrm{e}}^{-\mathrm{j}{\omega }_{k}t}‖}_2^2}\right\},\\ \mathrm{s}.\mathrm{t}.\hspace{1em}\hspace{1em}\hspace{1em}{\displaystyle \sum _k{u}_k\left(t\right)=f\left(t\right)},\end{array}\right.$$

(1)

where ${\omega }_k$ represents the center frequency of the kth mode. The constrained variational formulation presented in Equation (1) is resolved through the incorporation of two parameters: a penalty factor C and a Lagrangian multiplication operator $\theta \left(t\right)$. This transformation converts the originally constrained optimization problem into an unconstrained equivalent problem:

$$\begin{array}{ll}\hfill \mathrm{L}\left(\left\{u_k\right\},\left\{{\omega }_k\right\},\theta \right)=& C{\displaystyle \sum _k{‖{\partial }_t\left[\left(\delta \left(t\right)+\frac{j}{\pi t}\right)u_k\left(t\right)\right]{\mathrm{e}}^{-j{\omega }_{k}t}‖}_2^2}\hfill \\ & +{‖f\left(t\right)-{\displaystyle \sum _k{u}_k}‖}_2^2+〈\theta \left(t\right),f\left(t\right)-{\displaystyle \sum _k{u}_k\left(t\right)}〉,\hfill \end{array}$$

(2)

where ${‖f(t)-{\displaystyle \sum _k{u}_k\left(t\right)}‖}_2^2$ is the second penalty term, and $〈\cdot 〉$ represents the inner product operation.

The resolution of the unconstrained variational Equation (2) is achieved through the alternating direction method of multipliers, where the optimization parameters $u_k\left(t\right)$, ${\omega }_k$ and $\theta \left(t\right)$ are alternately updated as follows:

$${\widehat{u}}_k^{n+1}\left(\omega \right)={\displaystyle \frac{\widehat{f}\left(\omega \right)-{\displaystyle \sum _{i\ne k}{\widehat{u}}_i\left(\omega \right)+\left(\widehat{\theta }\left(\omega \right)/2\right)}}{1+2C{\left(\omega -{\omega }_k\right)}^2}},$$

(3)

$${\omega }_k^{n+1}={\displaystyle \frac{{\displaystyle {\int }_0^{\infty }\omega {\left|{\widehat{u}}_k\left(\omega \right)\right|}^{2}d\omega }}{{\displaystyle {\int }_0^{\infty }{\left|{\widehat{u}}_k\left(\omega \right)\right|}^{2}d\omega }}},$$

(4)

$${\widehat{\theta }}^{n+1}\left(\omega \right)={\widehat{\theta }}^n\left(\omega \right)+\tau \left[\widehat{f}\left(\omega \right)-{\displaystyle \sum _k{\widehat{u}}_k^{n+1}\left(\omega \right)}\right].$$

(5)

Performing an inverse Fourier transform on ${\widehat{u}}_k\left(\omega \right)$, the real part is the solution $\left\{u_k\left(t\right)\right\}\left(k=1,2,\dots ,K\right)$. By employing an iterative optimization framework to resolve the constrained variational formulation, VMD effectively separates the input signal into distinct quasi-orthogonal modes. During the procedure, VMD performs adaptive frequency-domain adjustments to each component, enabling reasonable spectral segmentation, which finally produces K narrowband modes.

2.2. Echo State Network

ESN serves as a dynamical system approximator in the VMD-ESN model, learning and predicting the temporal evolution of decomposed aerodynamic noise modes by leveraging its high-dimensional reservoir states to resolve temporal correlations in the time series. As shown in Figure 1, the ESN architecture comprises three fundamental components: an input layer, a high-dimensional processing reservoir, and an output layer. The network operates by projecting the input vector $u(t)$ into a higher-dimensional state space, creating a rich set of features that capture the temporal dynamics. Supposing that the dynamic state of the reservoir at a discrete time $t(t=0,\Delta t,2\Delta t,\mathrm{\dots })$ is characterized by a state vector $r(t)\in {\mathit{R}}^{N\times 1}$, the state update of the ESN reservoir is described by the equation below:

$$r(t+\Delta t)=f(A\cdot r(t)+W_{\mathrm{in}}\cdot u(t)),$$

(6)

where f is the activation function, A is an adjacency matrix which reflects the connection relationships between elements within the vector $r(t)$ and ${\mathit{W}}_{\mathrm{in}}$ is the mapping matrix from the input state to the reservoir state space. The output vector $\widehat{v}(t)$ can be obtained from

$$\widehat{v}(t)=f_{\mathrm{out}}(W_{\mathrm{out}},r(t)),$$

(7)

where ${\mathit{W}}_{\mathrm{out}}$ is the mapping matrix from the reservoir state space to the output state, and $f_{\mathrm{out}}$ is an element-wise output function. Generally, using the ridge regression algorithm can improve the stability of numerical calculations, and then obtain the output connection matrix ${\mathit{W}}_{\mathrm{out}}$, so that the output vector $v(t)$ approximates the desired output as closely as possible. During the training phase for a time period $-T\le t<0$, ${\mathit{W}}_{\mathrm{out}}$ is optimized via ridge regression to minimize the prediction error:

$${\widehat{\mathit{W}}}_{\mathrm{out}}=\underset{{\mathit{W}}_{\mathrm{out}}}{\arg \min }\left[{\displaystyle \sum _{-T\le t<0}{‖f_{\mathrm{out}}({\mathit{W}}_{\mathrm{out}},\mathit{r}(t))-v(t)‖}^2+\beta {‖{\mathit{W}}_{\mathrm{out}}‖}^2}\right],$$

(8)

where $\beta >0$ represents the regularization parameter. The solution of Equation (8) can be expressed in matrix form:

$${\widehat{W}}_{\mathrm{out}}=V{R}^{\mathrm{T}}{(R{R}^{\mathrm{T}}+\beta I)}^{-1},$$

(9)

where $R=(r(-T),r(-T+\Delta t),\mathrm{\dots },r(-\Delta t))$, $V=(v(-T),v(-T+\Delta t),\mathrm{\dots },v(-\Delta t))$, $v(t)$ is the desired output, and I is the identity matrix. During the prediction phase (t > 0), the output v(t) is computed through Equation (7). Since only the output weights need to be trained, the computational complexity is significantly reduced compared to other recurrent networks, which allows for less training time.

2.3. Synthesis of Predicted Subseries in VMD-ESN Model

As shown in Figure 1, the VMD-ESN model synthesizes predictions by aggregating the predicted subseries of all decomposed modes through linear superposition. Low-frequency modes carry the fundamental vortex-shedding energy and govern macroscopic pressure evolution. Their spectrally concentrated characteristics align with ESN’s memory capability, enabling accurate modeling with fewer neurons. High-frequency modes capture turbulent microscale fluctuations governing transient peaks, requiring sparser reservoir connectivity to resolve transient features. Each decomposed mode undergoes rolling prediction by ESN individually, and the final result is obtained by summing all predicted subseries, a process that harnesses the collective predictive power of the ensemble model. This aggregation not only enhances the robustness of the prediction by mitigating the inherent impact of noise in each subseries but also leverages the unique information captured by different VMD modes. This synthesis preserves spectral fidelity while balancing local mode accuracy and global signal consistency, which is critical for aerodynamic noise under high-Reynolds-number flows where energy spans broad frequency ranges. The combined result integrates VMD’s adaptive decomposition with ESN’s nonlinear dynamics modeling, enabling accurate predictions of transient noise behavior.

3. Time Series Prediction of Aerodynamic Noise

3.1. Data Preparation

Aerodynamic noise induced by flow around a cylinder is a critical factor in engineering design and serves as the foundation for techniques of noise control. Here, the sound pressure time series of aerodynamic noise induced by flow around a cylinder is taken as the research object.

Referring to the experimental model conducted by Revell et al. [33], we employ ANSYS Fluent computational fluid dynamics (CFD) software with launcher version of 18.0.0 to perform numerical simulations of aerodynamic noise induced by flow around a circular cylinder, generating the dataset for model training and prediction. As is illustrated in Figure 2, a two-dimensional physical model of aerodynamic noise induced by flow around a cylinder is established. The cylinder has a diameter of 19 mm. On the left side, the velocity inlet is defined, while the right side is specified as the pressure outlet. The top and bottom sides are set as symmetry boundaries. The Smagorinsky–Lilly model of large eddy simulation is used, and the fluid medium is an incompressible ideal gas at a temperature of 300 K with a viscosity of 1.7894 × 10⁻⁵ kg/(m·s). The time series of aerodynamic noise is captured by a receiver positioned at (0, −0.665 m). Figure 3 shows sound pressure values collected in 3988 steps with inlet velocities of 60 m/s, 69.2 m/s, and 80 m/s. Among these cases, the one with an inlet velocity of 69.2 m/s and a Reynolds number of 90,000 is a classic case and has been widely used to validate and compare the performance of different methods and turbulence models. The high Reynolds number means it is at a turbulent state where the inertial forces significantly outweigh the viscous forces and result in a complex and unstable flow behavior, which poses significant challenges and provides an ideal benchmark for testing the accuracy and reliability of turbulence prediction models.

3.2. Time Series Prediction Based on VMD-ESN Model

The time series is decomposed into 300 modes using VMD, as empirical testing reveals that this count optimally separates distinct physical processes without introducing artificial mode splitting or excessive computational burden, and the decomposition accuracy is set to $\epsilon =1\times {10}^{-7} \mathrm{Pa}$. A subset of the modes obtained from VMD is shown in Figure 4. Within a small time interval, each mode can be regarded as a harmonic signal, which proves that the modes are narrowband signals with energy concentrated around the center frequency. The time series to be predicted becomes clearer and stationary after the decomposition, with noise isolated and system redundancy reduced.

The training data is set to the aerodynamic noise sound pressure time series around the circular cylinder in the first 3500 time steps obtained from the numerical simulation, and through rolling prediction, the networks will predict the next 300 steps after the 3500th step. To reduce the influence of noise of the reservoir in the initial state, data in the first 500 time steps is selected as a hot start to initialize the reservoir. In order to compare the prediction effect of the VMD-ESN prediction model, ESN, EMD-ESN, and CEEMDAN-ESN will be used to predict the aerodynamic noise time series under identical conditions. The traditional ESN prediction model incorporates 2000 neurons, while other prediction models utilize 800 neurons in their ESN reservoirs. All configurations maintain identical topological parameters, with the spectral radius set to 0.7 and the average degree set to 3. The spectral radius of 0.7 ensures echo state property while maintaining sufficient memory capacity, and the average degree of 3 provides appropriate network connectivity. The configuration of the computer is Intel^® Core™ i7-4790 CPU operating at 3.60 GHz.

In the VMD-ESN method, sound pressure values of the first 3500 time steps of each subseries decomposed by VMD are used as the input vector to predict the value of the 3501st time step. When the first iteration is performed, the predicted output vector is substituted into the ESN as a new input vector for the next prediction epoch. Namely, the predicted sound pressure value at the 3501st time step, along with the preceding 3499 time steps, is then recombined into a new input vector. This updated input vector consists of the sound pressure values from the second time step to the 3501st time step. This process is repeated until predictions for all the set time steps are completed. All the prediction results are added up to obtain the final prediction result of the aerodynamic noise time series of flow around a cylinder. This rolling prediction strategy simulates the evolution process of the sound pressure time series of aerodynamic noise around a cylinder and maximizes the use of all available data for learning.

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. Prediction performance comparison of different models.

Time Steps Ahead	Model	RMSE	R2
50	ESN	5.1514	−0.5559
	EMD-ESN	4.6260	0.6395
	CEEMDAN-ESN	11.1393	−1.0905
	VMD-ESN	1.0524	0.9813
150	ESN	15.1145	−30.7351
	EMD-ESN	11.0795	0.0497
	CEEMDAN-ESN	9.3568	0.3223
	VMD-ESN	3.3225	0.9145

. To comprehensively assess model accuracy, two distinct metrics are implemented: the root-mean-square-error (RMSE) and the determination coefficient (R²):

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{{\displaystyle \sum _{i=1}^N{[y(i)-y_{\mathrm{data}}(i)]}^2}}{N}}},$$

(10)

$${\mathrm{R}}^2=1-{\displaystyle \frac{{\displaystyle \sum _{i=1}^N{[y(i)-y_{\mathrm{data}}(i)]}^2}}{{\displaystyle \sum _{i=1}^N{[y(i)-\overline{y}]}^2}}},$$

(11)

where N is the number of predicted time steps, $y(i)$ denotes the value at the ith time step in the predicted series, $y_{\mathrm{data}}(i)$ is the actual value at the ith time step, and $\overline{y}$ is the average value of the time series. In Equation (9), as R² 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 R² 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 R² 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 R² 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 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 ≈ 78,000) and 80 m/s (Re ≈ 104,000) are tested. Aerodynamic noise prediction performance is detailed in Figure 6 and

Table 2. Prediction performance of the VMD-ESN model.

Inflow Velocity	Time Steps Ahead	RMSE	R2
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

. At the lower Reynolds number (Re ≈ 78,000), 250 decomposition modes outperform a 300-mode configuration, confirming VMD’s adaptability to flow energy variations. Within short-term predictions of 50 steps, both cases achieve R² above 0.98 and low RMSE. For prediction within 150 steps, all cases maintain R² above 0.95 with RMSE below 3.96 dB, confirming temporal stability under varying flow conditions. The model maintains high accuracy at Re ≈ 78,000 and 104,000, validating robust generalization capacity for flow energy variations and turbulence characteristics in practical aerodynamic systems.

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 Kármán 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.