  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.
