Origin-Destination (OD) flow data is an important instrument in transportation studies. Precise prediction of customer demands from each original location to a destination given a series of previous snapshots helps  ride-sharing platforms to better understand their market mechanism. However, most existing prediction methods ignore the network structure of OD flow data and fail to utilize the topological dependencies among related OD pairs. In this paper, we propose a latent spatial-temporal origin-destination (LSTOD) model, with a novel convolutional neural network (CNN) filter to learn the spatial features of OD pairs from a graph perspective and an attention structure to capture their long-term periodicity. Experiments on a real customer request dataset with available OD information from a ride-sharing platform demonstrate the advantage of LSTOD in achieving at least 6.5% improvement in prediction accuracy over the second best model. Spatial-temporal prediction of large-scale network-based OD flow data plays an important role in traffic flow control, urban routes planning, infrastructure construction, and the policy design of ridesharing platforms, among others. On ride-sharing platforms, customers keep sending requests with origins and destinations at each moment. Knowing the exact original location and destination of each future trip allows platforms to prepare sufficient supplies in advance to optimize resource utilization and improve users' experience. Given the destinations of prospective demands, platforms can predict the number of drivers transferring from busy to idle status. Prediction of dynamic demand flow data helps ride-sharing platforms to design better order dispatch and fleet management policies for achieving the demand-supply equilibrium as well as decreased passenger waiting times and increased driver serving rates. Many efforts have been devoted to developing traffic flow prediction models in the past few decades. Before the rise of deep learning, traditional statistical and machine learning approaches dominate this field (Li et al., 2012; Lippi et al., 2013; Moreira-Matias et al., 2013; Shekhar & Williams, 2007; Idé & Sugiyama, 2011; Zheng & Ni, 2013) . Most of these models are linear and thus ignore some important non-linear correlations among the OD flows. Some other methods (Kwon & Murphy, 2000; Yang et al., 2013) further use additional manually extracted external features, but they fail to automatically extract the spatial representation of OD data. Moreover, they roughly combine the spatial and temporal features when fitting the prediction model instead of dynamically modelling their interactions. The development of deep learning technologies brings a significant improvement of OD flow prediction by extracting non-linear latent structures that cannot be easily covered by feature engineering. (Xingjian et al., 2015; Ke et al., 2017; Zhou et al., 2018) . Zhang et al. (2016; modeled the whole city are as an entire image and employed residual neural network to capture temporal closeness. and also learned traffic as images but they used LSTM instead to obtain the temporal dependency. Yao et al. (2018b) proposed a Deep Multi-View Spatial-Temporal Network (DMVST-Net) framework to model both spatial and temporal relationships. However, using standard convolution filters suffers from the problem that some OD flows covered by a receptive field of regular CNNs are not spatially important. Graph-based neural net-works (GNN) (Kipf & Welling, 2016; Defferrard et al., 2016; Veličković et al., 2017) are proved to be powerful tools in modelling spatial-temporal network structures Li et al., 2017) . However, none of these frameworks are directly applicable here since both the historical observations and responses to predict are vertex-level variables. On the contrary, the OD flows we discuss in this paper are generated in the edge space by our definition. The aim of this paper is to introduce a hierarchical Latent Spatial-Temporal Origin-Destination (LSTOD) prediction model to jointly extract the complex spatial-temporal features of OD data by using some well-designed CNN-based architectures. Instead of modelling the dynamic OD networks as a sequence of images and applying standard convolution filters to capture their spatial information, we introduce a novel Vertex Adjacent Convolution Network (VACN) that uses an irregular convolution filter to cover the most related OD flows that share common vertecies with the target one. The OD flows connected by common starting and/or ending vertexes, which may fall into different regions of the flow map, can be spatially correlated and topologically connected. Moreover, for most ride-sharing platforms, a passenger is more likely to send a new request from the location where his/her last trip ends in. To learn such sequential dependency, we introduce a temporal gated CNN (TGCNN) and integrate it with VACN by using the sandwich-structured STconv block in order to collectively catch the evolutionary mechanism of dynamic OD flow systems. A periodically shifted attention mechanism is also used to capture the shift in the long-term periodicity. Finally, the combined short-term and long-term representations are fed into the final prediction layer to complete the architecture. Our contributions are summarized as follow: • To the best of our knowledge, it is the first time that we propose purely convolutional structures to learn both short-term and long-term spatio-temporal features simultaneously from dynamic origin-destination flow data. • We propose a novel VACN architecture to capture the graph-based semantic connections and functional similarities among correlated OD flows by modeling each OD flow map as an adjacency matrix. • We design a periodically shift attention mechanism to model the long-term periodicity when using convolutional architecture TGCNN in learning temporal features. • Experimental results on two real customer demand data sets obtained from a ride-sharing platform demonstrate that LSTOD outperforms many state-of-the-art methods in OD flow prediction, with 7.94% to 15.14% improvement of testing RMSE.
We propose a purely convolutional CNN model with attention mechanism to predict spatial-temporal origin-destination flows. 
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