Traffic flow forecasting is a fundamental research issue for transportation planning and management, which serves as a canonical and typical example of spatial-temporal predictions. In recent years, Graph Neural Networks (GNNs) and Recurrent Neural Networks (RNNs) have achieved great success in capturing spatial-temporal correlations for traffic flow forecasting. Yet, two non-ignorable issues haven't been well solved: 1) The message passing in GNNs is immediate, while in reality the spatial message interactions among neighboring nodes can be delayed. The change of traffic flow at one node will take several minutes, i.e., time delay, to influence its connected neighbors. 2) Traffic conditions undergo continuous changes. The prediction frequency for traffic flow forecasting may vary based on specific scenario requirements. Most existing discretized models require retraining for each prediction horizon, restricting their applicability. To tackle the above issues, we propose a neural Spatial-Temporal Delay Differential Equation model, namely STDDE. It includes both delay effects and continuity into a unified delay differential equation framework, which explicitly models the time delay in spatial information propagation. Furthermore, theoretical proofs are provided to show its stability. Then we design a learnable traffic-graph time-delay estimator, which utilizes the continuity of the hidden states to achieve the gradient backward process. Finally, we propose a continuous output module, allowing us to accurately predict traffic flow at various frequencies, which provides more flexibility and adaptability to different scenarios. Extensive experiments show the superiority of the proposed STDDE along with competitive computational efficiency.

翻译：交通流预测是交通规划与管理的基础研究问题，也是时空预测的典型范例。近年来，图神经网络（GNNs）和循环神经网络（RNNs）在捕捉交通流预测中的时空相关性方面取得了显著成功。然而，两个不可忽视的问题尚未得到妥善解决：1）GNNs中的消息传递是瞬时的，而现实中相邻节点间的空间消息交互可能存在延迟。一个节点的交通流变化需要几分钟（即时间延迟）才能影响其连接的相邻节点。2）交通状况处于持续变化中。交通流预测的频率可能因具体场景需求而异。现有的大多数离散化模型需要针对每个预测周期重新训练，限制了其适用性。为解决上述问题，我们提出一种神经时空延迟微分方程模型，即STDDE。它通过统一的延迟微分方程框架同时纳入延迟效应和连续性，明确建模了空间信息传播中的时间延迟。此外，我们提供了理论证明来展示其稳定性。随后，我们设计了一个可学习的交通图时延估计器，利用隐藏状态的连续性实现梯度反向过程。最后，我们提出一个连续输出模块，能够以不同频率准确预测交通流，从而为不同场景提供了更高的灵活性和适应性。大量实验表明，所提出的STDDE具有优越性以及颇具竞争力的计算效率。