The document presents a novel spatio-temporal graph neural controlled differential equation (stg-ncde) model aimed at enhancing traffic forecasting accuracy by integrating neural controlled differential equations with graph convolutional networks. Experimental results demonstrated that stg-ncde outperforms various baseline models across multiple datasets, showcasing its capability to handle irregular time-series data without changing its architecture. The authors suggest that this combination of technologies offers promising directions for future research in spatio-temporal processing.

1. Graph Neural Controlled Differential Equations for
Traffic Forecasting
Jeongwhan Choi Hwangyong Choi
Jeehyun Hwang Noseong Park
Yonsei University
Seoul, South Korea
AAAI 2022
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2. Table of Contents
1 Introduction
2 Proposed Method
3 Experiments
4 Conclusion
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3. Table of Contents
1 Introduction
2 Proposed Method
3 Experiments
4 Conclusion
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4. Traffic Forecasting (1/2)
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5. Traffic Forecasting (2/2)
Predicting traffic flows in a city is of great importance to Intelligence
Traffic Systems (ITS):
help manage and control traffic to ease congestion,
pre-allocate taxis and sharing bikes,
route planning.
Figure: Example of heavy traffic (credit: KTLA)
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6. Spatio-temporal Forecasting (1/2)
Figure: Traffic congestion in front of the Yonsei AI workshop venue
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7. Spatio-temporal Forecasting (2/2)
The traffic forecasting task is one of the most popular problems in the
area of spatio-temporal processing.
Spatio-temporal forecasting is to predict the traffic volume for each
location of a road network for the next time points given past
historical traffic pattern.
......
Figure: Example of spatio-temporal graph data
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8. Motivation (1/3)
Neural Controlled Differential Equations
(NCDEs) are considered as a
continuous analogue to recurrent neural
networks (RNNs).
NCDEs show the state-of-the-art
accuracy in many time-series tasks.
Time
Path
NCDE
Data
Hidden
Neural CDE
......
......
......
Figure: The overall workflow
of the original NCDE for
processing time-series
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9. Motivation (2/3)
Model Temporal Dependency Spatial Dependency
STGCN [11] GLU GCN
DCRNN [8] LSTM GCN (Pre-defined Graph)
GraphWaveNet [10] TCN GCN (Adaptive Graph)
ASTGCN [5] 1D CNN+Attention Attention+GCN(Pre-defined Graph)
STSGCN [9] GCN (Sliding Window) GCN (Multiple Graph)
AGCRN [1] GRU GCN (Adaptive Graph)
STFGNN [7] 1D CNN GCN (Multiple Graph)
STGODE [4] TCN ODE+GCN (Dynamic Graph)
STG-NDCE NCDE NCDE + GCN (Adaptive Graph)
Table: Classification of spatial-temporal modeling methods
Motivation
It has not been studied yet how to combine the NCDE technology and the
graph convolutional processing technology.
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10. Motivation (3/3)
Objective
We integrate them into a single framework to solve the spatio-temporal
forecasting problem.
Time
Path
NCDE
Data
Hidden
Neural CDE
......
......
......
Figure: The overall workflow of the
original NCDE for processing time-series
Time
Path
NCDE
Data
Hidden
Temporal Processing
Spatial Processing
NCDE
Hidden
......
......
......
......
......
Figure: The overall workflow in our
proposed STG-NCDE
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11. Table of Contents
1 Introduction
2 Proposed Method
3 Experiments
4 Conclusion
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12. Graph Neural Controlled Differential Equations
Our proposed spatio-temporal graph neural controlled differential
equation (STG-NCDE) consists of two NCDEs.
Time
Path
NCDE
Data
Hidden
Temporal Processing
Spatial Processing
NCDE
Hidden
......
......
......
......
......
Figure: The overall workflow in our proposed STG-NCDE.
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13. Table of Contents
1 Introduction
2 Proposed Method
3 Experiments
4 Conclusion
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14. Experimental Results (1/2)
Model
PeMSD3 PeMSD4 PeMSD7 PeMSD8
MAE RMSE MAPE MAE RMSE MAPE MAE RMSE MAPE MAE RMSE MAPE
HA 31.58 52.39 33.78% 38.03 59.24 27.88% 45.12 65.64 24.51% 34.86 59.24 27.88%
ARIMA 35.41 47.59 33.78% 33.73 48.80 24.18% 38.17 59.27 19.46% 31.09 44.32 22.73%
VAR 23.65 38.26 24.51% 24.54 38.61 17.24% 50.22 75.63 32.22% 19.19 29.81 13.10%
FC-LSTM 21.33 35.11 23.33% 26.77 40.65 18.23% 29.98 45.94 13.20% 23.09 35.17 14.99%
TCN 19.32 33.55 19.93% 23.22 37.26 15.59% 32.72 42.23 14.26% 22.72 35.79 14.03%
TCN(w/o causal) 18.87 32.24 18.63% 22.81 36.87 14.31% 30.53 41.02 13.88% 21.42 34.03 13.09%
GRU-ED 19.12 32.85 19.31% 23.68 39.27 16.44% 27.66 43.49 12.20% 22.00 36.22 13.33%
DSANet 21.29 34.55 23.21% 22.79 35.77 16.03% 31.36 49.11 14.43% 17.14 26.96 11.32%
STGCN 17.55 30.42 17.34% 21.16 34.89 13.83% 25.33 39.34 11.21% 17.50 27.09 11.29%
DCRNN 17.99 30.31 18.34% 21.22 33.44 14.17% 25.22 38.61 11.82% 16.82 26.36 10.92%
GraphWaveNet 19.12 32.77 18.89% 24.89 39.66 17.29% 26.39 41.50 11.97% 18.28 30.05 12.15%
ASTGCN(r) 17.34 29.56 17.21% 22.93 35.22 16.56% 24.01 37.87 10.73% 18.25 28.06 11.64%
MSTGCN 19.54 31.93 23.86% 23.96 37.21 14.33% 29.00 43.73 14.30% 19.00 29.15 12.38%
STG2Seq 19.03 29.83 21.55% 25.20 38.48 18.77% 32.77 47.16 20.16% 20.17 30.71 17.32%
LSGCN 17.94 29.85 16.98% 21.53 33.86 13.18% 27.31 41.46 11.98% 17.73 26.76 11.20%
STSGCN 17.48 29.21 16.78% 21.19 33.65 13.90% 24.26 39.03 10.21% 17.13 26.80 10.96%
AGCRN 15.98 28.25 15.23% 19.83 32.26 12.97% 22.37 36.55 9.12% 15.95 25.22 10.09%
STFGNN 16.77 28.34 16.30% 20.48 32.51 16.77% 23.46 36.60 9.21% 16.94 26.25 10.60%
STGODE 16.50 27.84 16.69% 20.84 32.82 13.77% 22.59 37.54 10.14% 16.81 25.97 10.62%
Z-GCNETs 16.64 28.15 16.39% 19.50 31.61 12.78% 21.77 35.17 9.25% 15.76 25.11 10.01%
STG-NCDE 15.57 27.09 15.06% 19.21 31.09 12.76% 20.53 33.84 8.80% 15.45 24.81 9.92%
Table: Forecasting error on PeMSD3, PeMSD4, PeMSD7 and PeMSD8
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15. Experimental Results (2/2)
06:00 12:00 18:00 24:00 06:00
Time
100
200
300
400
Traffic
Flow
Truth
STG-NCDE
Z-GCNETs
(a) Node 261 in PeMSD4
12:00 18:00 24:00 06:00 12:00
Time
100
200
300
400
500
600
700
800
Traffic
Flow
Truth
STG-NCDE
Z-GCNETs
(b) Node 9 in PeMSD8
Figure: Traffic forecasting visualization.
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16. Irregular traffic forecasting
Since NCDEs are able to consider irregular time-series by the design,
STG-NCDE is also able to do it without any changes on its model
design.
Table: Forecasting error on irregular PeMSD4
Model Missing rate MAE RMSE MAPE
STG-NCDE
10% 19.36 31.28 12.79%
30% 19.40 31.30 13.04%
50% 19.98 32.09 13.48%
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17. Table of Contents
1 Introduction
2 Proposed Method
3 Experiments
4 Conclusion
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18. Conclusion
Time
Path
NCDE
Data
Hidden
Temporal Processing
Spatial Processing
NCDE
Hidden
......
......
......
......
......
Figure: STG-NCDE
We presented a spatio-temporal NCDE
model to perform traffic forecasting.
In our experiments with 6 datasets and
20 baselines, our method clearly shows
the best overall accuracy.
In addition, our model can perform
irregular traffic forecasting.
We believe that the combination of
NCDEs and GCNs is a promising
research direction for spatio-temporal
processing.
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19. Reference (1/3)
[1] Lei Bai et al. “Adaptive Graph Convolutional Recurrent Network for
Traffic Forecasting”. In: NeurIPS. Vol. 33. 2020, pp. 17804–17815.
[2] Ricky T. Q. Chen et al. “Neural Ordinary Differential Equations”.
In: NeurIPS. 2018.
[3] Yuzhou Chen, Ignacio Segovia-Dominguez, and Yulia R Gel.
“Z-GCNETs: Time Zigzags at Graph Convolutional Networks for
Time Series Forecasting”. In: ICML. 2021.
[4] Zheng Fang et al. “Spatial-Temporal Graph ODE Networks for
Traffic Flow Forecasting”. In: KDD. 2021.
[5] Shengnan Guo et al. “Attention Based Spatial-Temporal Graph
Convolutional Networks for Traffic Flow Forecasting”. In: AAAI. July
2019. doi: 10.1609/aaai.v33i01.3301922.
[6] Patrick Kidger et al. “Neural Controlled Differential Equations for
Irregular Time Series”. In: NeurIPS (2020).
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20. Reference (2/3)
[7] Mengzhang Li and Zhanxing Zhu. “Spatial-Temporal Fusion Graph
Neural Networks for Traffic Flow Forecasting”. In: AAAI. May 2021.
[8] Yaguang Li et al. “Diffusion Convolutional Recurrent Neural
Network: Data-Driven Traffic Forecasting”. In: ICLR. 2018.
[9] Chao Song et al. “Spatial-Temporal Synchronous Graph
Convolutional Networks: A New Framework for Spatial-Temporal
Network Data Forecasting”. In: AAAI. Apr. 2020. doi:
10.1609/aaai.v34i01.5438.
[10] Zonghan Wu et al. “Graph WaveNet for Deep Spatial-Temporal
Graph Modeling”. In: IJCAI. July 2019, pp. 1907–1913.
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21. Reference (3/3)
[11] Bing Yu, Haoteng Yin, and Zhanxing Zhu. “Spatio-Temporal Graph
Convolutional Networks: A Deep Learning Framework for Traffic
Forecasting”. In: IJCAI. July 2018. doi:
10.24963/ijcai.2018/505. url:
https://doi.org/10.24963/ijcai.2018/505.
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22. Thank You!
Visit our poster session for more details.
homepage: jeongwhanchoi.me
email: jeongwhan.choi@yonsei.ac.kr
github: jeongwhanchoi/STG-NCDE
twitter: jeongwhan_choi
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23. More Details
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24. Neural Ordinary Differential Equations (NODEs)
NODEs introduced how to
continuously model residual neural
networks (ResNets) with differential
equations.
NODE can be written as follows:
z(T) = z(0) +
Z T
0
f (z(t), t; θf )dt; (1)
where the neural network
parameterized by θf approximates
z(t)
dt .
Figure: ResNet and NODE figures
from [2]
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25. Neural controlled differential equations (NCDEs)
NCDEs [6] generalize RNNs in a continuous
manner and can be written as follows:
z(T) = z(0) +
Z T
0
f (z(t); θf )dX(t) (2)
= z(0) +
Z T
0
f (z(t); θf )
dX(t)
dt
dt, (3)
Time
Path
NCDE
Data
Hidden
Neural CDE
......
......
......
Figure: The overall workflow
of the original NCDE for
processing time-series.
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26. Neural controlled differential equations (NCDEs)
NCDEs [6] generalize RNNs in a continuous
manner and can be written as follows:
z(T) = z(0) +
Z T
0
f (z(t); θf )dX(t) (2)
= z(0) +
Z T
0
f (z(t); θf )
dX(t)
dt
dt, (3)
The original CDE formulation in Eq. (2)
reduces Eq. (3) where z(t)
dt is approximated
by f (z(t); θf )dX(t)
dt .
Time
Path
NCDE
Data
Hidden
Neural CDE
......
......
......
Figure: The overall workflow
of the original NCDE for
processing time-series.
Jeongwhan Choi (Yonsei Univ.) STG-NCDE Yonsei AI Workshop 2022 25 / 34

27. Graph Neural Controlled Differential Equations (1/7)
Initial value generation
Our proposed spatio-temporal graph neural controlled differential
equation (STG-NCDE) consists of two NCDEs.
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28. Graph Neural Controlled Differential Equations (2/7)
Initial value generation
Time
Path
NCDE
Data
Hidden
Temporal Processing
Spatial Processing
NCDE
Hidden
......
......
......
......
......
Figure: The overall workflow in our proposed STG-NCDE.
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29. Graph Neural Controlled Differential Equations (3/7)
Initial value generation
Time
Path
NCDE
Data
Hidden
Temporal Processing
Spatial Processing
NCDE
Hidden
......
......
......
......
......
Figure: STG-NCDE
The first NCDE for the temporal
processing can be written as follows:
H(T) = H(0) +
Z T
0
f (H(t); θf )
dX(t)
dt
dt,
(4)
where H(v)(t) is a hidden trajectory
(over time t ∈ [0, T]) of the temporal
information of node v, and X(t) is a
matrix whose v-th row is X(v).
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30. Graph Neural Controlled Differential Equations (4/7)
Initial value generation
Time
Path
NCDE
Data
Hidden
Temporal Processing
Spatial Processing
NCDE
Hidden
......
......
......
......
......
Figure: STG-NCDE
The second NCDE starts for its spatial
processing as follows:
Z(T) = Z(0) +
Z T
0
g(Z(t); θg )
dH(t)
dt
dt,
(5)
where the hidden trajectory Z(t) is
controlled.
After combining Eqs. (4) and (5), we
have the following single equation
which incorporates both the temporal
and the spatial processing:
Z(T) = Z(0) +
Z T
0
g(Z(t); θg )f (H(t); θf )
dX(t)
dt
dt
(6)
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31. Graph Neural Controlled Differential Equations (5/7)
Initial value generation
The definition of f : R|V|×dim(h(v)) → R|V|×dim(h(v)) is as as follows:
f (H(t); θf ) = ψ(FC|V|×dim(h(v))→|V|×dim(h(v))(AK )),
.
.
.
A1 = σ(FC|V|×dim(h(v))→|V|×dim(h(v))(A0)),
A0 = σ(FC|V|×dim(h(v))→|V|×dim(h(v))(H(t))),
(7)
where σ is a rectified linear unit and ψ is a hyperbolic tangent.
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32. Graph Neural Controlled Differential Equations (6/7)
Initial value generation
The definition of g : R|V|×dim(z(v)) → R|V|×dim(z(v)) is as follows:
g(Z(t); θg ) = ψ(FC|V|×dim(z(v))→|V|×dim(z(v))(B1)), (8)
B1 = (I + ϕ(σ(E · E⊺
)))B0Wspatial , (9)
B0 = σ(FC|V|×dim(z(v))→|V|×dim(z(v))(Z(t))), (10)
where I is the |V| × |V| identity matrix, ϕ is a softmax activation,
E ∈ R|V|×C is a trainable node-embedding matrix and Wspatial is a
trainable weight transformation matrix.
ϕ(σ(E · E⊺)) corresponds to the normalized adjacency matrix
D−1
2 AD−1
2 , where A = σ(E · E⊺) and the softmax activation plays a
role of normalizing the adaptive adjacency matrix [10, 1].
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33. Graph Neural Controlled Differential Equations (7/7)
Initial value generation
The initial value of the temporal processing as follows:
H(0) = FCD→dim(h(v))(Ft0 ).
We also use the following similar strategy to generate Z(0):
Z(0) = FCdim(h(v))→dim(z(v))(H(0)).
After generating these initial values for the two NCDEs, we can
calculate Z(T) after solving the Riemann–Stieltjes integral problem.
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34. Datasets
We use six real-world traffic datasets which were widely used in the
previous studies [11, 5, 4, 3, 9].
Dataset |V| Time Steps Time Range Type
PeMSD3 358 26,208 09/2018 - 11/2018 Volume
PeMSD4 307 16,992 01/2018 - 02/2018 Volume
PeMSD7 883 28,224 05/2017 - 08/2017 Volume
PeMSD8 170 17,856 07/2016 - 08/2016 Volume
PeMSD7(M) 228 12,672 05/2012 - 06/2012 Velocity
PeMSD7(L) 1,026 12,672 05/2012 - 06/2012 Velocity
Table: The summary of the datasets used in our work. We predict either traffic
volume (i.e., # of vehicles) or velocity.
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35. Experimental Settings
We use the forecasting settings of S = 12 and M = 1 after reading
past 12 graph snapshots.
We use the mean absolute error (MAE), the mean absolute
percentage error (MAPE), and the root mean squared error (RMSE)
to measure the performance.
We compare our proposed STG-NCDE with the 20 baseline models.
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