## 2 Related Work  
In the realm of spatial-temporal series prediction, the two primary categories of methods are statistical methods and machine learning methods. Statistical prediction methods such as vector autoregressive (VAR) [6], Kalman filtering model (KM) [7], and k-nearest neighbor (KNN) [8] can be used for the development of suitable mathematical models. However, they are limited in their capacity to extract dynamic and fixed spatial dependencies. As an upgraded version of autoregressive (AR) [12], the VAR achieves spatial-temporal series prediction by constructing multiple equations to calculate the dynamic relationship of all endogenous variables. However, the VAR model's limitations are evident in its inability to effectively predict spatial-temporal series due to it cannot leverage the dynamic dependencies among multiple variables. Anyway, statistical prediction methods often encounter challenges in making long-term predictions and are unable to capture dynamic spatial-temporal dependencies. To address the weakness of these statistical prediction methods, traditional shallow machine learning methods, such as support vector machine (SVM) [9] and random forest [10], can better capture inherent laws but not for multivariate prediction because they emphasize nonlinear mapping relationships rather than sequence connections. As highly complex systems, recurrent neural networks (RNNs) [13] and their variations, including long short-term memory (LSTM) [14] and gated recurrent unit (GRU) [15], are capable of capturing temporal correlations. The diffusion convolutional recurrent neural network (DCRNN) [16] is a model that integrates diffusion convolution and recurrent neural networks (RNNs) to capture spatial-temporal dependencies to make a prediction. [PERSON] et al. [17] proposed the CNN-LSTM model by fusing the LSTM and convolutional neural network (CNN), achieving great performance in short-term prediction. [PERSON] et al. [18] proposed multiple CNN models for periodic multivariate time prediction. Nevertheless, limited by convolution kernel size, it is difficult to capture long-term temporal dependencies with CNNs. The proposal of a self-attention mechanism was verified as a powerful tool for long-term prediction. Spatial-temporal transformer networks (STTNs) [19] adopts self-attention to model the temporal dependencies and achieve competitive results. The attention-based spatial-temporal graph neural network (ASTGNN) [20] embeds CNN in self-attention to enable temporal dynamics and obtain global receptive fields that significantly improve long-term prediction accuracy. LogTrans [21] adopts local convolution and LogSparse attention to reduce the complexity to \(O(L(logL)^{2})\). Informer [22] achieves complexity \(O(LogL)\) by introducing KL-divergence-based ProbSparse attention. Autoformer [23] presents series decomposition and utilizes a custom-designed Auto-Correlation mechanism that dramatically improves long-term prediction accuracy. Spatial dependencies modeling is also a significant part of spatial-temporal prediction. The deep multi-view spatial-temporal network (DMVST-Net) [24] applies local CNNs to capture the local characteristics of regions about the neighbors of each node. However, the CNNs [25] cannot be adaptive to graph-based spatial data. Subsequent to the emergence of graph neural networks (GNNs) [26], graph convolution networks such as ChebNet [27] were proposed later. The temporal graph convolutional network (T-GCN) [28] extracts spatial-temporal dependencies simultaneously through combining the GCN [29] with GRU. Spatio-temporal graph convolutional networks (STGCN) [30] develops a complete convolutional structure composed of the GCN and CNN that replaces convolutional and recurrent units and outperformed the baselines. STTNs introduces ChebNet to extract fixed spatial dependencies. Nevertheless, it is challenging for GCNs to deal with hidden and dynamic spatial dependencies. GraphWaveNet [31] learns a self-adaptive adjacent matrix for hidden spatial dependencies.