Coarse-to-Fine Learning of Dynamic Causal Structures

Dezhi Yang; Qiaoyu Tan; Carlotta Domeniconi; Jun Wang; Lizhen Cui; Guoxian Yu

cs.LGFeb 26, 2026

Coarse-to-Fine Learning of Dynamic Causal Structures

Authors: Dezhi Yang, Qiaoyu Tan, Carlotta Domeniconi, Jun Wang, Lizhen Cui, Guoxian Yu

Organizations: School of Software Shandong University Jinan, China · Department of Computer Science New York University Shanghai, China · Department of Computer Science George Mason University VA, USA · Shandong Institutes of Industrial Technology Jinan, China

Abstract

Learning the dynamic causal structure of time series is a challenging problem. Most existing approaches rely on distributional or structural invariance to uncover underlying causal dynamics, assuming stationary or partially stationary causality. However, these assumptions often conflict with the complex, time-varying causal relationships observed in real-world systems. This motivates the need for methods that address fully dynamic causality, where both instantaneous and lagged dependencies evolve over time. Such a setting poses significant challenges for the efficiency and stability of causal discovery. To address these challenges, we introduce DyCausal, a dynamic causal structure learning framework. DyCausal leverages convolutional networks to capture causal patterns within coarse-grained time windows, and then applies linear interpolation to refine causal structures at each time step, thereby recovering fine-grained and time-varying causal graphs. In addition, we propose an acyclic constraint based on matrix norm scaling, which improves efficiency while effectively constraining loops in evolving causal structures. Comprehensive evaluations on both synthetic and real-world datasets demonstrate that DyCausal achieves superior performance compared to existing methods, offering a stable and efficient approach for identifying fully dynamic causal structures from coarse to fine.

Figures & tables

Figure 1: Conceptual overview of DyCausal . DyCausal uses a sliding convolutional window to traverse temporal series and encodes coarse-grained time-varying causal structures. It then defines a linear interpolation strategy inspired by Taylor expansion to refine the causal structures at each time step

\{\mathbf{W}_{t}\}^{T}_{t=1}

. Based on the acyclic constraint

h_{norm}

enforced by matrix

1

-norm, DyCausal reconstructs time series to optimize causal structures and obtains dynamic causal graphs

\mathcal{G}

that conform to the dynamic distribution of time series.

Figure 2: The decoding process. Vector

\mathbf{Z}_{t}

is cut and decoded in parallel to obtain vectors

\{\tilde{\mathbf{z}}_{t,i}\}^{m}_{i=1}

, which are again cut, recombined and decoded in parallel to obtain vectors

\{\tilde{\mathbf{u}}_{t,j}\in\mathbb{R}^{dm}\}^{d\tau}_{j=1}

. The combination of

\{\tilde{\mathbf{u}}_{t,j}\in\mathbb{R}^{dm}\}^{d\tau}_{j=1}

forms the matrix

\mathbf{W}_{t}

Panel of Figure 3 — Figure 3: Results on time series with dynamic causality. $\{\mathbf{W}_{1},\cdots,\mathbf{W}_{10}\}$ represent the causal graphs estimated at each time step.

Figure 4: Comparison of acyclic constraints and their gradients, and runtime.

Methods	Traffic		AQI		Medical
Methods	Precision $\uparrow$	F1 $\uparrow$	Precision $\uparrow$	F1 $\uparrow$	Precision $\uparrow$	F1 $\uparrow$
DyCausal	87.41 $\pm$ 3.85	54.81 $\pm$ 2.51	75.31 $\pm$ 1.43	63.16 $\pm$ 1.23	83.21 $\pm$ 2.75	56.86 $\pm$ 1.35
CUTS+	80.10 $\pm$ 3.16	45.21 $\pm$ 1.74	65.60 $\pm$ 3.47	58.20 $\pm$ 2.44	44.82 $\pm$ 3.54	30.23 $\pm$ 1.85
JRNGC	68.04 $\pm$ 5.56	52.04 $\pm$ 2.41	72.34 $\pm$ 3.43	55.36 $\pm$ 1.61	55.56 $\pm$ 0.63	52.05 $\pm$ 0.75
NTS-NO	67.43 $\pm$ 5.35	46.07 $\pm$ 1.85	72.70 $\pm$ 2.99	40.09 $\pm$ 1.29	44.80 $\pm$ 2.13	34.27 $\pm$ 1.89
DYNO	47.63 $\pm$ 3.26	45.34 $\pm$ 2.92	62.55 $\pm$ 1.34	44.60 $\pm$ 0.75	48.40 $\pm$ 2.07	31.70 $\pm$ 1.25

Table 1: Results on CausalTime. The boldfaced and underlined results respectively highlight the methods that perform the first and second best among all compared methods. We omit the percentage signs

\%

of Precision and F1 to simplify the representation.

Figure 5: Estimated dynamic causal structures on traffic subset. We observed that: (i) causality

x_{6}\rightarrow x_{4}

x_{4}\rightarrow x_{19}

x_{6}\rightarrow x_{19}

x_{15}\rightarrow x_{7}

and

x_{19}\rightarrow x_{6}

are dynamic up to the time

t=20

and static after;(ii) causality

x_{4}\rightarrow x_{8}

x_{19}\rightarrow x_{4}

and

x_{16}\rightarrow x_{10}

are static before and after

t=20

, but with different causal strengths. In particular, the strength of

x_{4}\rightarrow x_{8}

is almost zero before

t=20

, which means that

x_{4}\rightarrow x_{8}

can only be identified from time series after

t=20

Appendix figures & tables20 assets

Supplementary material from the paper’s appendix.

Appendix

$t$	Methods	20 nodes			80 nodes
$t$	Methods	TPR $\uparrow$	F1 $\uparrow$	SHD $\downarrow$	TPR $\uparrow$	F1 $\uparrow$	SHD $\downarrow$
$\mathbf{W}_{1}$	DyCausal	89.54 $\pm$ 6.98	91.43 $\pm$ 5.60	7.40 $\pm$ 5.39	89.10 $\pm$ 1.70	93.85 $\pm$ 1.31	18.20 $\pm$ 3.87
	DYCAST	25.98 $\pm$ 10.33	29.51 $\pm$ 10.67	45.43 $\pm$ 7.21	-	-	-
	DYNO	3.59 $\pm$ 3.18	3.52 $\pm$ 2.90	78.40 $\pm$ 10.75	0.35 $\pm$ 0.70	0.34 $\pm$ 0.68	294.90 $\pm$ 28.51
	NTS-NO	3.89 $\pm$ 1.57	4.03 $\pm$ 1.69	72.33 $\pm$ 12.04	0.77 $\pm$ 1.01	0.78 $\pm$ 1.02	294.20 $\pm$ 15.55
$\mathbf{W}_{25}$	DyCausal	90.93 $\pm$ 6.01	94.38 $\pm$ 3.90	4.10 $\pm$ 2.88	91.01 $\pm$ 1.97	95.20 $\pm$ 1.18	13.40 $\pm$ 2.15

Appendix

Table A1: Results on time series with dynamic causality.

\mathbf{W}_{1}

and

\mathbf{W}_{50}

represent the causal graphs estimated at both ends of the time series.

\mathbf{W}_{25}

represents the causal graph estimated in the middle of the time series. We omit the percentage signs

\%

of TPR and F1 to simplify the representation.

Figure A1: Results on dynamic settings with

d=10

variables and

\tau=1

maximum lag.

\mathbf{W}_{1}

and

\mathbf{W}_{50}

represent the causal graphs estimated at both ends of the time series.

\mathbf{W}_{25}

represents the causal graph estimated in the middle of the time series. The left and right parts represent the ground-truth and the estimated causal matrices. Red and blue represent the positive and negative matrix weights respectively, and the darker the color, the greater the absolute value of the weight. The estimated matrices are basically consistent with the ground-truth matrices.

$t$	Methods	10 nodes			20 nodes
$t$	Methods	TPR $\uparrow$	F1 $\uparrow$	SHD $\downarrow$	TPR $\uparrow$	F1 $\uparrow$	SHD $\downarrow$
$\mathbf{W}_{1}$	DyCausal	90.35 $\pm$ 6.62	91.42 $\pm$ 4.39	3.20 $\pm$ 1.40	89.86 $\pm$ 3.83	92.26 $\pm$ 2.98	6.00 $\pm$ 2.37
$\mathbf{W}_{1}$	w/o inter	75.67 $\pm$ 22.20	72.98 $\pm$ 23.78	12.50 $\pm$ 13.86	90.06 $\pm$ 5.45	92.18 $\pm$ 4.12	6.10 $\pm$ 3.67
$\mathbf{W}_{25}$	DyCausal	90.41 $\pm$ 7.88	92.65 $\pm$ 7.73	3.10 $\pm$ 3.24	91.09 $\pm$ 5.06	93.64 $\pm$ 5.47	4.90 $\pm$ 4.57
$\mathbf{W}_{25}$	w/o inter	81.66 $\pm$ 15.05	83.25 $\pm$ 20.30	7.50 $\pm$ 11.15	88.69 $\pm$ 10.05	91.28 $\pm$ 9.88	6.40 $\pm$ 7.30
$\mathbf{W}_{50}$	DyCausal	89.62 $\pm$ 6.94	90.81 $\pm$ 3.67	3.50 $\pm$ 1.43	87.75 $\pm$ 6.53	89.40 $\pm$ 4.84	8.30 $\pm$ 4.03

Appendix

Table A2: Results on time series with dynamic causality.

\mathbf{W}_{1}

and

\mathbf{W}_{50}

represent the causal graphs estimated at both ends of the time series.

\mathbf{W}_{25}

represents the causal graph estimated in the middle of the time series. We omit the percentage signs

\%

of TPR and F1 to simplify the representation.

Figure A2: Results of DyCausal on datasets with number of nodes

d=\{50,100,200,300\}

Figure A3: Results of DyCausal on datasets with fewer samples

N=\{20,50,100,150\}

Figure A4: Results of DyCausal on datasets with times series lengths

T=\{10,30,50,70,90\}

Figure A5: Results of DyCausal on datasets with noise strength

\{0.5,1.0,2.5,5.0\}

Figure A6: Results of DyCausal on datasets with maximum lag

\tau=\{1,2,3,4,5\}

Figure A7: Results of DyCausal on datasets with maximum lag

\tau=3

Panel of Figure 16 — Figure A8: Results on time series with static settings.

linear structural equation model
	matrices	DyCausal	DYNO	PCMCI	SVAM
10 nodes	TPR $\uparrow$	97.13 $\pm$ 2.42	85.46 $\pm$ 6.95	95.59 $\pm$ 1.83	39.79 $\pm$ 22.34
	F1 $\uparrow$	98.45 $\pm$ 1.29	91.54 $\pm$ 4.45	58.87 $\pm$ 4.26	46.86 $\pm$ 24.15
	SHD $\downarrow$	1.80 $\pm$ 1.54	9.30 $\pm$ 4.65	88.30 $\pm$ 12.81	52.60 $\pm$ 19.27
20 nodes	TPR $\uparrow$	97.11 $\pm$ 2.43	93.38 $\pm$ 3.79	96.26 $\pm$ 1.67	65.35 $\pm$ 6.84
	F1 $\uparrow$	98.52 $\pm$ 2.43	96.54 $\pm$ 2.06	49.37 $\pm$ 4.94	70.40 $\pm$ 5.55

Appendix

Table A3: Results on time series with static causality. The boldfaced and underlined results respectively highlight the methods that perform the first and second best among all compared methods. We omit the percentage signs

\%

of TPR and F1 to simplify the representation.

Figure A9: Results of DyCausal and DyCausal with

h_{log}

. The left is the ground-truth causal graph, the middle is the causal graph estimated by DyCausal , and the right is the causal graph estimated by DyCausal with

h_{log}

Figure A10: The trajectory of acyclic constraint values indexed by the number of iterations.

Figure A11: Results of DyCausal and DyCausal with

h_{log}

. The left is the ground-truth causal graphs, the middle is the causal graphs estimated by DyCausal , and the right is the causal graphs estimated by DyCausal with

h_{log}

Figure A12: The trajectory of acyclic constraint values indexed by the number of iterations.

	NetSim ( $d=15$ )	DREAM ( $d=100$ )	Phoenix ( $d=34$ )	CausalRivers ( $d=5$ )
DyCausal	0.9391 $\pm$ 0.0075	0.6806 $\pm$ 0.0197	0.5472 $\pm$ 0.0185	0.7214 $\pm$ 0.0405
CUTS+	0.7058 $\pm$ 0.0071	0.6229 $\pm$ 0.0138	0.5104 $\pm$ 0.0151	0.6072 $\pm$ 0.0627
NGM	0.7499 $\pm$ 0.0020	0.5629 $\pm$ 0.0020	0.4809 $\pm$ 0.0083	0.4905 $\pm$ 0.0526

Appendix

Table A4: AUROC on NetSim, DREAM-3 and Phoenix.

		$\beta=0.001$	$\beta=0.005$	$\beta=0.01$	$\beta=0.05$	$\beta=0.1$
$\mathbf{W}_{1}$	TPR	86.07	88.91	77.09	93.65	55.95
	F1	66.46	77.36	70.02	95.49	63.68
	SHD	99.0	51.5	70.0	9.0	55.0
$\mathbf{W}_{25}$	TPR	92.87	95.78	98.95	98.93	79.06
	F1	88.37	94.77	98.43	99.46	86.83
	SHD	24.0	10.0	3.0	1.0	63.0

Appendix

Table A5: Results of sensitivity analysis on

\beta

	$P=4,E=2$			$P=1,E=2$
	$\mathbf{W}_{1}$	$\mathbf{W}_{4}$	$\mathbf{W}_{8}$	$\mathbf{W}_{1}$	$\mathbf{W}_{2}$	$\mathbf{W}_{3}$	$\mathbf{W}_{4}$	$\mathbf{W}_{5}$	$\mathbf{W}_{6}$	$\mathbf{W}_{7}$	$\mathbf{W}_{8}$	$\mathbf{W}_{9}$	$\mathbf{W}_{10}$
TPR	93.11	87.92	91.21	91.92	97.06	86.87	94.12	86.87	91.18	87.88	90.20	89.90	95.10
F1	92.78	97.21	90.93	88.78	92.52	82.30	90.57	83.90	88.15	85.29	88.46	84.76	89.81
SHD	13.7	5.3	17.2	23.00	16.00	37.00	20.00	33.00	25.00	30.00	24.00	32.00	22.00

Appendix

Table A6: Results on non-smooth dynamic causality.

	DyCausal			$K=4,S=1$			$K=1,S=1$
	TPR	F1	SHD	TPR	F1	SHD	TPR	F1	SHD
$\mathbf{W}_{1}$	91.56	92.73	63.0	90.54	90.95	72.0	83.33	20.48	2485.0
$\mathbf{W}_{25}$	96.70	98.32	13.0	95.47	97.68	18.0	89.22	30.22	1644.0
$\mathbf{W}_{50}$	94.18	94.18	46.0	90.09	92.59	64.0	85.68	20.76	2512.0
runtime (ms)	2926.0			3527.0			1970.0

Appendix

Table A7: Results for settings

K=4

S=1

and

K=1

S=1

results of instantaneous causal graphs
	$T=50,d=50$			$T=10,d=20$
	$\mathbf{W}_{1}$	$\mathbf{W}_{25}$	$\mathbf{W}_{50}$	$\mathbf{W}_{1}$	$\mathbf{W}_{2}$	$\mathbf{W}_{3}$	$\mathbf{W}_{4}$	$\mathbf{W}_{5}$	$\mathbf{W}_{6}$	$\mathbf{W}_{7}$	$\mathbf{W}_{8}$	$\mathbf{W}_{9}$	$\mathbf{W}_{10}$
TPR	92.03	97.34	91.52	86.91	83.03	86.12	81.39	91.29	91.29	81.56	85.19	83.60	83.64
F1	94.45	98.24	93.84	89.27	89.53	91.80	89.43	93.62	93.88	89.03	90.19	89.59	86.47
SHD	7.2	2.2	6.8	6.2	9.8	8.5	8.7	2.7	2.8	8.8	9.5	10.3	8.0
results of lagged causal graphs

Appendix

Table A8: Results on instantaneous and lagged causality.

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