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

TL;DR

STDDE tackles the core challenges of traffic forecasting by explicitly modeling spatial delays in information propagation and enabling continuous-time, horizon-adaptive predictions. It introduces delay-aware neural differential equations with a learnable traffic-graph time-delay estimator and a continuous output module, underpinned by a stability analysis showing asymptotic stability when $c \le 1/K$. Empirically, STDDE achieves state-of-the-art results on six real-world datasets and demonstrates robustness to varying input intervals and delays, while maintaining competitive efficiency. This work broadens ITS forecasting capabilities by bridging delayed spatial interactions with continuous-time dynamics, offering flexible, accurate predictions for diverse operational needs.

Abstract

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.

Figures (7)

• Figure 1: (a) and (b) show the comparison of spatial-temporal signal propagation between general GNNs and realistic conditions. Node 2 and node 4 receive the same update information simultaneously in graph propagation, while they do not in the realistic scene. Fig. (c) shows the distribution of delay values in the real-world traffic network, which are computed based on the max-cross-correlation method azaria1984time.
• Figure 2: The workflow comparison of original discrete time-series processing and CDE processing scheme.
• Figure 3: Overview of STGDDE. It consists of two components. The first component includes both delay effects and continuity into a unified delay differential equation framework, which explicitly models the time delay in spatial information propagation. The second component is the continuous output module, allowing us to accurately predict traffic flow at various frequencies.
• Figure 4: Comparison of prediction results between our model and STDDE-no-delay.
• Figure 5: Performance comparison with input time intervals greater than inference intervals.

Theorems & Definitions (10)

• Definition 1
• Theorem 1
• Definition 2
• Lemma 1
• Definition 3
• Theorem 2
• Definition 4
• Lemma 2
• Lemma 3
• Theorem 3