## 3 Time Series Prediction of Aerodynamic Noise
### 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 \(\varepsilon=1\times 10^{-7}\) 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.  
Figure 3: The aerodynamic noise sound pressure time series at the receiver with inlet velocity of (**a**) 60 m/s, (**b**) 69.2 m/s, (**c**) 80 m/s.  
Figure 2: The two-dimensional physical model of aerodynamic noise induced by flow around a cylinder.  
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 3500 th 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(r) Core(tm) 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 3501 st 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 3501 st 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 3501 st 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.  
Figure 4: VMD of the aerodynamic noise signal.