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.

提出一种神经时空延迟微分方程模型（STDDE）进行交通流预测，该模型结合了延迟效应和连续性，通过明确地建模时空信息传播中的时间延迟，设计了可学习的交通图时延估计器，并提出了连续输出模块，从而实现在不同频率下准确预测交通流的灵活性和适应性。

交通预测中的延迟效应揭示：来自时空延迟微分方程的视角

Unveiling Delay Effects in Traffic Forecasting: A Perspective from  Spatial-Temporal Delay Differential Equations

We propose a novel learning framework based on neural mean-field dynamics for
inference and estimation problems of diffusion on networks. Our new framework
is derived from the Mori-Zwanzig formalism to obtain an exact evolution of the
node infection probabilities, which renders a delay differential equation with
memory integral approximated by learnable time convolution operators, resulting
in a highly structured and interpretable RNN. Directly using cascade data, our
framework can jointly learn the structure of the diffusion network and the
evolution of infection probabilities, which are cornerstone to important
downstream applications such as influence maximization. Connections between
parameter learning and optimal control are also established. Empirical study
shows that our approach is versatile and robust to variations of the underlying
diffusion network models, and significantly outperform existing approaches in
accuracy and efficiency on both synthetic and real-world data.

本文提出了一种基于神经平均场动力学的新型学习框架，用于解决网络扩散的推理和估计问题，该框架利用 Mori-Zwanzig 形式主义从节点感染概率的精确演化获得，形 成一个高度结构化和可解释的 RNN，可用于联合学习扩散网络的结构和感染概率的演化，这是影响最大化等重要下游应用的基石。此外，本文还建立了参数学习和最优控制之间的联系。实证研究表明，我们的方法具有多样性和鲁棒性，可以在合成数据和真实世界数据上显著优于现有方法的准确性和效率。