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

arXiv: https://arxiv.org/abs/2402.01231 · HTML: https://mlsanity.com/paper/2402.01231/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 51 references that are in the map.
- First posted on arXiv 2 Feb 2024.

## Builds on (most cited first, 18 of 18)

- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Spatio-Temporal Graph Convolutional Networks: A Deep Learning Framework for Traffic Forecasting](https://mlsanity.com/paper/1709.04875/) (2017)
- [Adaptive Graph Convolutional Recurrent Network for Traffic Forecasting](https://mlsanity.com/paper/2007.02842/) (2020)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Deep Graph Library: A Graph-Centric, Highly-Performant Package for Graph Neural Networks](https://mlsanity.com/paper/1909.01315/) (2019)
- [GRU-ODE-Bayes: Continuous modeling of sporadically-observed time series](https://mlsanity.com/paper/1905.12374/) (2019)
- [Spatial-Temporal Graph ODE Networks for Traffic Flow Forecasting](https://mlsanity.com/paper/2106.12931/) (2021)
- [Wider or Deeper: Revisiting the ResNet Model for Visual Recognition](https://mlsanity.com/paper/1611.10080/) (2016)
- [Graph Neural Controlled Differential Equations for Traffic Forecasting](https://mlsanity.com/paper/2112.03558/) (2021)
- [Deep Learning for Time Series Forecasting: Tutorial and Literature Survey](https://mlsanity.com/paper/2004.10240/) (2020)
- [Learning Long-Term Dependencies in Irregularly-Sampled Time Series](https://mlsanity.com/paper/2006.04418/) (2020)
- [A Comprehensive Survey on Deep Graph Representation Learning](https://mlsanity.com/paper/2304.05055/) (2023)
- [Neural Delay Differential Equations](https://mlsanity.com/paper/2102.10801/) (2021)
- [Theoretically Improving Graph Neural Networks via Anonymous Walk Graph Kernels](https://mlsanity.com/paper/2104.02995/) (2021)
- [A Survey of Data-Efficient Graph Learning](https://mlsanity.com/paper/2402.00447/) (2024)
- [Graph Structural-topic Neural Network](https://mlsanity.com/paper/2006.14278/) (2020)

## Built on by (most cited first, 11 of 11)

- [Patch-wise Structural Loss for Time Series Forecasting](https://mlsanity.com/paper/2503.00877/) (2025)
- [PPGF: Probability Pattern-Guided Time Series Forecasting](https://mlsanity.com/paper/2502.12802/) (2025)
- [Graph ODEs and Beyond: A Comprehensive Survey on Integrating Differential Equations with Graph Neural Networks](https://mlsanity.com/paper/2503.23167/) (2025)
- [GUME: Graphs and User Modalities Enhancement for Long-Tail Multimodal Recommendation](https://mlsanity.com/paper/2407.12338/) (2024)
- [Addressing Prediction Delays in Time Series Forecasting: A Continuous GRU Approach with Derivative Regularization](https://mlsanity.com/paper/2407.01622/) (2024)
- [Damba-ST: Domain-Adaptive Mamba for Efficient Urban Spatio-Temporal Prediction](https://mlsanity.com/paper/2506.18939/) (2025)
- [MillGNN: Learning Multi-Scale Lead-Lag Dependencies for Multi-Variate Time Series Forecasting](https://mlsanity.com/paper/2509.03852/) (2025)
- [Prior Knowledge-enhanced Spatio-temporal Epidemic Forecasting](https://mlsanity.com/paper/2602.22270/) (2026)
- [Efficient Traffic State Prediction With Dynamic Joint Spatio-Temporal Relation Inference](https://mlsanity.com/paper/2504.08061/) (2025)
- [C3DE: Causal-Aware Collaborative Neural Controlled Differential Equation for Long-Term Urban Crowd Flow Prediction](https://mlsanity.com/paper/2509.12289/) (2025)
- [DeMa: Dual-Path Delay-Aware Mamba for Efficient Multivariate Time Series Analysis](https://mlsanity.com/paper/2601.05527/) (2026)

## Abstract (arXiv)

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.