## 1 Introduction
While EWT is more versatile due to its ability to customize the transformation based on signal characteristics, VMD was selected in this study because it offers better control over the decomposition of specific frequency components. This makes it highly effective for handling complex and noisy subway passenger flow data, where the accurate separation of frequency components is critical [19; 20; 21; 22; 23]. Overcoming the shortcomings of current methods in the research process poses a significant challenge in the area of subway passenger flow forecasting. The latest research trend involves the effective improvement and proper integration of mathematical statistics methods, machine learning, and deep learning approaches, exploiting the advantages of each model while avoiding their risks and drawbacks [24]. The primary benefit of enhancing diverse models for subway passenger flow data is their targeted approach; they can analyze and forecast from various perspectives, considering the \"non-stationary\" and \"seasonal\" attributes of subway passenger flow data [25; 26; 27; 28; 29]. For instance, the SARIMA model and TimesNet model, although they are mathematical statistics and deep learning models, respectively, can model time series targeting the \"seasonality\", \"smoothness\", and \"intra-period\" and \"inter-period\" characteristics of passenger flow data. In light of the above models' characteristics, this paper proposes an improved TimesNet subway passenger flow prediction model based on Multi-scale Variational Mode Decomposition (VMD) and the incorporation of a seasonal module. The model decomposes subway passenger flow data via VMD into IMFs of various frequencies, then feeds these decomposed IMFs into the improved TimesNet model with a seasonal module for prediction and, finally, reconstructs the predicted outcome. In our proposed method, judging from the seasonal detection module, the SARIMA component is primarily responsible for capturing the linear trends and seasonal variations in subway passenger flow data, while the improved TimesNet parallel is tasked with capturing and simulating the nonlinear patterns and complex \"within-period\" and \"across-period\" dependencies of subway passenger flow data [30]. Such targeted improvements, tailored to the characteristics of subway passenger flow data, can effectively enhance the accuracy of subway passenger flow predictions. Meanwhile, it is important to clarify that the \"seasonal variations\" referred to in this study specifically address short-term seasonality. In the context of subway passenger flow data, this refers to recurring patterns observed within a single day, such as the morning and evening rush hours, as well as the differences between weekdays and weekends. These short-term cycles are critical for understanding and optimizing subway operations, as they provide valuable insights into passenger flow fluctuations throughout the day and week. The experiments conducted in this paper selected real swipe card data from the Hangzhou subway in 2019 (a total of 70 million records) and processed them to produce subway passenger flow data with a time step of five minutes. In time series analysis, especially for complex urban transportation systems, it is crucial to account for both linear and nonlinear dynamics. While linear models such as SARIMA are effective at capturing the seasonal and trend components of the data, many real-world systems, including metro passenger flow, exhibit nonlinear and chaotic behavior. Nonlinear dynamics can be identified through various metrics such as the Lyapunov exponent, correlation dimension, and pseudo-attractor analysis [31; 32]. In this paper, we verify the nonlinearity and chaotic nature of the dataset time series by analyzing them as described above. The results demonstrate that our method shows significant performance improvements over other baseline models (e.g., Autoformer, TimesNet, Crossformer) in terms of the Mean Absolute Error (MAE) and Mean Squared Error (MSE) evaluation metrics.