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

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 STDDE. Both quantitative and qualitative experiments are conducted to validate the concept of a delay-aware module. Also, the flexibility validation shows the effectiveness of the continuous output module.