## 1 Introduction  
With the rapid growth of the urban population, immense traffic pressure has inevitably been imposed on transportation systems. Compared to other transportation modes, rail transit, with its efficiency and high passenger capacity, is integral to sustainable development. As the rail transit network expands and urban land remains limited, metro stations have become larger and deeper to alleviate traffic congestion. This has resulted in significant passenger flows, especially during peak hours. Passenger flow is a fundamental measurement reflecting the crowding level in the increasingly extensive and complex metro network system. Accurately and effectively predicting passenger flow enhances overall metro operational efficiency, alleviates peak-hour congestion, and reduces operating costs. Therefore, as a crucial component of intelligent transportation systems, scientifically forecasting passenger flow holds essential practical significance [1]. Passenger flow data refers to time series data consecutively recorded within a specific time interval. The analysis of passenger flow data is the core of metro passenger flow prediction, helping to find the changes in passenger flow and providing the scientific basis for prediction. In recent decades, there has been a sustained emphasis on time series prediction in academic research, with widespread applications across diverse fields such as financial market prediction [2], climate prediction [3], and traffic prediction [4; 5]. Existing studies can be generally categorized into two classes: traditional methods and deep learning methods. Traditional methods include statistical analysis and machine learning approaches. Statistical analysis-based methods often treat passenger flow data as a time series prediction problem. For instance, the Autoregressive Integrated Moving Average (ARIMA) model, developed from autoregressive models, is one of the most typical time series forecasting models. Refs. [6; 7; 8] proposed a framework for analyzing short-term passenger flow during the special event based on the ARIMA model, demonstrating its superior accuracy and reliability. In addition to passenger flow, ARIMA is widely employed for stock price forecasting [9] and traffic prediction [10]. While statistical forecasting models such as ARIMA have proven effective in certain contexts, they often struggle with the nonlinear capabilities [11] inherent in metro passenger flow data. Machine learning methods provide a more flexible, data-driven alternative for learning temporal dynamics [12]. Traditional machine learning models, such as K-Nearest Neighbors [13], Support Vector Machine [14], and neural network models [15], are trained on historical data through supervised learning for predictive tasks. While these models can capture specific nonlinear structures within metro passenger flow data, they still face limitations in feature engineering and accuracy in complex scenarios. Deep learning, however, has emerged as a powerful tool, capable of leveraging potent nonlinear modeling capabilities, capturing more abstract latent features from extensive datasets, and reducing the need for manual feature engineering and model design [16]. Over the past few years, deep learning methods have made remarkable breakthroughs [17]. In studies of traffic flow or passenger flow using neural networks, previous studies are generally based on Multilayer Perceptron (MLP) and their variants [18; 19]. It generates intermediate hidden layer feature representations through a series of non-linear layers to ensure ease of implementation. Yet, their accuracy tends to decrease when handling high-dimensional complex data [20]. Recurrent neural network (RNN) is another deep learning model suitable for processing sequential data. Unlike Multilayer Perceptron, RNN units integrate current data and memory capabilities. However, retaining long-term memory often leads to vanishing and exploding gradient problems, making it difficult to capture long-term dependencies [21].