[Paper Review] Graph Neural Controlled Differential Equations for Traffic Forecasting

Jeongwhan Choi, Hwangyong Choi+2 more|arXiv (Cornell University)|Dec 7, 2021

Traffic Prediction and Management Techniques28 citations

TL;DR

The paper introduces STG-NCDE, a spatio-temporal forecasting model combining two neural controlled differential equations (NCDEs) for temporal and spatial processing, showing state-of-the-art results across six PeMS datasets and robustness to irregular data.

ABSTRACT

Traffic forecasting is one of the most popular spatio-temporal tasks in the field of machine learning. A prevalent approach in the field is to combine graph convolutional networks and recurrent neural networks for the spatio-temporal processing. There has been fierce competition and many novel methods have been proposed. In this paper, we present the method of spatio-temporal graph neural controlled differential equation (STG-NCDE). Neural controlled differential equations (NCDEs) are a breakthrough concept for processing sequential data. We extend the concept and design two NCDEs: one for the temporal processing and the other for the spatial processing. After that, we combine them into a single framework. We conduct experiments with 6 benchmark datasets and 20 baselines. STG-NCDE shows the best accuracy in all cases, outperforming all those 20 baselines by non-trivial margins.

Motivation & Objective

• Motivate high-accuracy traffic forecasting on spatio-temporal graphs with irregular time-series data.
• Develop a dual NCDE framework to separately model temporal and spatial dependencies.
• Leverage adaptive graph convolution within NCDEs to capture dynamic spatial relations.
• Demonstrate robustness to missing observations and irregular sampling in real-world traffic data.
• Provide comprehensive large-scale evaluation against numerous baselines.

Proposed method

• Create a continuous-path representation X^(v) for each node v via natural cubic spline interpolation of past features.
• Two NCDEs are used: temporal processing h^(v)(T) driven by dX^(v)/dt (Eq. 4/5), and spatial processing Z(T) driven by dH/dt (Eq. 6).
• Define f as a multi-layer fully-connected, per-node temporal processor (Eq. 8).
• Define g to perform spatial mixing using an adaptive adjacency (E, W_spatial) and a softmax-normalized interaction (Eq. 9–11).
• Combine temporal and spatial NCDEs into a single augmented ODE (Eq. 12) and generate predictions ŷ^(v) from Z(T) (Eq. 13).
• Train with L1 loss plus weight decay; prove well-posedness via Lipschitz continuity considerations.

Experimental results

Research questions

• RQ1 Can a dual NCDE framework effectively capture both temporal dynamics and spatial dependencies in traffic networks?
• RQ2 Does integrating an adaptive, NCDE-based spatial module with a temporal NCDE improve forecasting accuracy over strong baselines?
• RQ3 How robust is STG-NCDE to irregular or missing sensor data in real-world traffic scenarios?
• RQ4 What is the impact of node embedding size and path interpolation on model performance?

Key findings

• Outperforms 20 baselines across 6 PeMS datasets on MAE, RMSE, and MAPE (STG-NCDE achieves the best average accuracy).
• STG-NCDE remains superior under irregular data scenarios with 10–50% missing rates, while baselines struggle.
• Ablations show that using both temporal and spatial NCDEs yields better results than either alone.
• Increasing node embedding size C up to 10 improves performance, with stability observed beyond C=7.
• The model provides precise predictions across horizons and demonstrates robust training dynamics versus ablations.