Coarse-to-Fine Learning of Dynamic Causal Structures
Authors: Dezhi Yang
Qiaoyu Tan
Carlotta Domeniconi
Jun Wang
Lizhen Cui
Guoxian Yu
Venue: ICLR 2026 Poster (2026)
Keywords: causal discovery
dynamic causality
coarse-to-fine
matrix norm scaling
Primary area: causal reasoning
TL;DR: A causal discovery framework that learns dynamic causal graphs from coarse to fine with a theoretically and empirically stable acyclic constraint.

Abstract: Learning the dynamic causal structure is a difficult challenge in discovering causality from time series. Most existing studies rely on distributional or structural invariance to uncover the underlying causal dynamics, assuming stationary or partially stationary causality, which frequently conflicts with complex causal relationships in the real world. This boosts temporal causal discovery to encompass fully dynamic causality, where both instantaneous and lagged causal dependencies may change over time, bringing significant challenges to the efficiency and stability of causal discovery. To tackle these challenges, we introduce DyCausal, a dynamic causal structure learning framework that leverages convolutional networks to effectively model causal structures within coarse-grained time windows, and introduces linear interpolation to refine causal structures to each time step and recover time-varying causal graphs. In addition, we propose an acyclic constraint based on matrix norm scaling. It is more stable both theoretically and empirically, and constrains loops in dynamic causal structures with improved efficiency. Evaluations on both synthetic and real-world datasets prove that DyCausal significantly outperforms existing methods and identifies fully dynamic causal structures from coarse to fine.

Published as a conference paper at ICLR 2026
COARSE-TO-FINE LEARNING OF DYNAMIC CAUSAL STRUCTURES
Dezhi Yang
Qiaoyu Tan
School Software
Department of Computer Science
Shandong University
New York University
Jinan, China
Shanghai, China
dzyang@mail sdu. edu. on
qiaoyu. tan tanrnyu. edu
Carlofta Domeniconi
Jun Wang
Department of Computer Science
Shandong Institutes of Industrial Technology Jinan, China
George Mason University
VA, USA
k kingjun edu,
carlottašcs, gmu edu
Lizhen Cui, Guoxian Yu*
School of Software
Shandong University
Jinan, China
{clz, gxyu)esdu.edu.co
ABSTRACT
Learning the dynamic causal structure of time series is 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 meth- ods that address fully dynamic causality, where both instantancous and lagged dependencies evolve over time. Such a setting poses significant challenges for the efficiency and stability of causal discovery. To address these challenges, in- troduce DyCausal, a dynamic causal structure learning framework. DyCausal leverages convolutional networks to capture causal patterns within grassegg time windows, and then applies linear interpolation to refine causal structures at each time step. thereby recovering fine- mee-graindd 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 stable and efficient approach for identifying fully dynamic causal structures from coarse to fine.
INTRODUCTION
Temporal causal relationships can decipher the dynamics of complex real-world processes and play critical roles in boosting scientific discoveries in wide range of disciplines, such as traffic (Cheng al. (0244b,, healthcare (Qian al., 2020), meteorology (Nowack al., (020), and finance (Ham- moudeh et al., 2020). Granger causality (Ganger 1969) serves as the cornerstone for temporal causal analysis; determines causality based on the predictability of the cause series to the effect series. Its high explainability and compatibility with deep models continue to attract research ded-
icated uncovering causal graphs from observed time series (Marinazzo et al. 2008 Khanna &
Tan 2019 Liu et al. 2024).
"Corresponding author

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length step
size S
step 2
step 3
Reconstruct Time Series
time series
reconstructed loss:
dynamic causal graphs
|IX -
acyelic constraint
Xg
log det(s/ Wil
deg
Sliding Convolutional Window
1+2
143
H+
1+5
116
6
++28
Linear Interpolation
1
(W.
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 { [W Based the acyclic constraint horm enforced by matrix 1-norm, DyCausa reconstructs time series to optimize causal structures and obtains dynamic causal graphs G that conform the dynamic distribution of time series.
Recognizing and modeling dynamic causality is crucial for unraveling the underlying causal mech- anisms in complex systems. Despite substantial progress in temporal causal discovery (Runge et al.,
2019
De Brouwer et 2020, Bellot et al., 2022 Tank al. 2021] Sun al. 2023 Zhou et al., 2024 Wu et al. [204]] Yao et al. 2025), these methods still focus on evolving time series with static causal models, thus cannot fulfill the demand of identifying dynamic causality in real-world applications. In dynamic causal settings, time series often exhibit changing distributions, with causal relationships emerging, changing, or vanishing over time (c.g., scasonal shifts in patient disease trajectories). To handle this, some recent methods (Fujiwara et al. 2023 Cheng et al.] 2025) impose static assumptions on certain causal components to model partial dy- namic causality. This often runs counter to the inherent dynamic nature of the real world. How to realize causal discovery towards more practical applications, faithfully modeling fully dynamic causality, where all causal relations can change over time, is urgently necessary but has not been well explored yet.
Pamfil et al.
2020
key Learning fully dynamic temporal causal structures is quite challenging, primarily due
obstacles: (i) efficient and stable discovery of time-varying causal graphs, given the complexity of dynamic signals; (ii) effective enforcement of acyclicity within causal graphs, especially when causal relationships are rapidly changing. The frequently changing causality at each time step causes expensive and tond-to-converge causal models. Fortunately, for causality that changes continuously over time, good identification of its coarse-grained states at a few time steps can, in principle, support approximating
complete time-varying trajectory. To this end, we propose a framework for coarse-to- corrse-toffine fone learning fully dynamic causal structures (DyCausal), Specifically, DyCausal leverages convolutional networks to encode causal structures within coarse- gurrse-rrined sliding win- dows, and then refines causal structures at each time step through linear interpolation. In addition, an acyclic constraint based on matrix norm scaling is introduced to optimize the causal model. By scaling the causal matrix an optimizable space, the constraint guides the model to identify causal graphs with improved stability and efficiency. Figure []] illustrates the conceptual framework of DyCausal. Our main contributions summarized as follows:
(i) DyCausal meticulously combines the sliding convolutional window and linear interpolation to efficiently capture the coarse-to-fin temporal evolution of causal structures. This design success- fully models fully dynamic causal graphs and boosts flexible adaption to various causal models, broadening its applicability diverse domains.
(ii) DyCausal defines an enhanced strategy for the log-determinant-basee acyclic constraint, uti- lizing the matrix norm to achieve an always differentiable and optimizable constraint. This strategy not only avoids retraining and hyperparameter search, but also improves the stability and efficiency of learning dynamic causal graphs.
(iii) Extensive experiments both synthetic and real-world datasets show that DyCausal consis- tently outperforms state-off of-thee-rt art methods, demonstrating its ability to capture fully dynamic causal evolution from coarse fine while enforcing theoretically stable acyelic constraint.
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2 RELATED WORK
Identifying causal graphs from time series has become an important topic in causal learning, with existing approaches broadly falling into two categories: constraint- bestraiitt-baeddd and score-based methods. Constraint-basee methods (Entner& & Hoyer 2010) extend the two classic algorithms, PC (Spirtes) & Glymour 199T) and FCI (Spirtes et al. 2000), time series and determine causal relationships by conditional independence tests. Ruugee et al. (201)); Gerhardus & Runge (2020) later combined these two algorithms with linear and nonlinear conditional independence tests and achieved scalabil- ity on large-scale time series. Score- use vector autoregressive loss and sparse regularization identity causal models that correctly re- construct time series. Inspired by NOTEARS (Zheng et al. 2018). Pamfil et al. (220); Sun et al.)
methods (Hyvärinen et al.) 2010 2008 TTak et al. (02T))
(2023) introduced the differentiable acyclic constraint to optimize autoregressive processes in linear and nonlinear models. Gao et al. (2022); Li et al. (2023) considered intervention data to enhance the identifiability of temporal causal structures. In addition, [ellott et al. (2022) used the ordinary dif- ferential equation to reconstruct time series. Many studies (Khanna & Tan 2019 De Brouwer al. 2020 Cheng et al. 2023 2024a), including the aforementioned ones, have made great progress, but all assume that the time series evolve from static causality and fail identify dynamic causality as shown in the experimental section, which limits their application in the real world.
We focus more realistic scenarios with dynamic causal relationships, causal discovery for such scenarios is still under-explored. In this context, [illig et al. (2025) introduced a meta-causal model capture dynamic causal relationships in the bivariate system. SSagiioro et al. (2020); Gao et al. (2023) divided time series into segments and identified the causal mechanism in each segment. Gaao al. (2023) modeled the causality smooth function of the time index. These methods focus on the dynamic data distribution, but are still limited to learning static causal structures. (Song et al.
y Gao & Yang (2022) utilized kernel-reweighted autoregression to model dynamic causal graphs. Hallac et al. (201)) developed message-passsng algorithm to infer time-varying structures. Cheng et al. (2025) used the ordinary differential equation to model the latent dynamics of causal structures. These methods introduce the dynamic to lagged instantaneous structures only, and none of them learns the fully dynamic causality. In contrast, our DyCausal leverages sliding convolutional windows and linear interpolation recover fully dynamic causality from coarse to fine, and a constraint based on the matrix norm to enforce acyclic causal graphs with improved efficiency and stability.
PROBLEM DEFINITION
Given N independent time series X' = (X i.T,.... .X of length T, the series n E [1 at time t E [1,7]] records d variable values X = {-/_,. [.a) With a slight abuse of notation, we denote the observed data at time as X, and ignore the serial index. Without loss of generality, we assume that all series have the same time-varying causal structures.
Structural equation model. We introduce the structural equation model (SEM) to formalize tem- poral causality, where the generation process of zτ, is defined follows:
Ft.i =
(It,5); P Atog(11.3) + 6:
where E; is the exogenous noise independent of the observed variables, PAins (It,s) and PAiag (II..) respectively represent the parents that have instantaneous and lagged effects on Ft,i. In general, SEM assumes maximum lag T.
Given generation functions J (fi. fa), we denote the causal structure as weighted adja- cency matrix composed of partial derivatives W (F)= = ... p E (0, r] and only if st-p.j is independent on Zt,5-
[afa] (..]....ffal Rarxd, wi + pd, = O.
We assume that there is cycle in instantaneous causal relationships. This is a key assumption in causal discovery, simplifying the difficulty of modeling causality and ensuring distin- guishable causes and effects. Unlike non-temporal causal discovery, temporal causal graphs contain both instantaneous and lagged causal relationships, where the causes of lagged causality clearly occur before the effects and do not involve acyclicity. By splitting the causal matrix W into the instantaneous matrix Wins and lagged one Wlog we only require that Wins be a directed acyclic graph, Wins E DAGs.
Acyclicity.
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m {2r,)
[ūr,)j=1 )
m [z,
rii
{ue
dr
iii
We
g-c
Zt
(MLP ym
[MLP])J=1
Figure 2: The decoding process. Vector Z is cut and decoded in parallel to obtain vectors {zt,a & hm i== which are again cut, recombined and decoded in parallel to obtain vectors {ür.j Rdm |dz combination of (ū.. (Rdmndd 1j=1 forms the matrix W
The
j==1
Dynamic causal structures. Given the ground-truth weighted adjacency matrix W., E [1, denote its time-varying trajectory as a smooth function of the time index as follows:
W. F(Wo. t), Wine DAGs
In this paper, are committed to learning the tome-varying functions F and extracting the weighted adjacency matrix W..
4 FULLY DYNAMIC CAUSAL STRUCTURE LEARNING
In this section, we introduce a coarse-to- coarse-to-fine strategy to encode the time-varying trajectory of causal structures, and propose efficient acyclic constraint integrated into the optimization process, sulting in a reliable framework for identifying fully dynamic causal graphs.
4.1 COARSE-TO-FINE ENCODING OF CAUSAL MATRIX
We propose mine the coarse grained states of dynamic causal structures across multiple time steps. Referring to Eq. (2). the causality is defined as continuously evolving over time, and the few coarse-grained states of dynamic causality can support approximating its fine-grained time-varying trajectory. We design sliding convolutional window of width N and length K, and slide it over the time series with step size S. At each sliding step, the convolutional network encodes the data within the window state vector:
Z, = C=CNN(Xx:tt
where X K are the data within the window, CNN((- the convolutional neural network, and Z, E dR embeds the data as coarse-grained state representation of causal structures. We defer the discussion of m Section 4.3 Next, we decode Z, onto weight adjacency matrix represent causal structures at time l.
We divide Z, into short vectors Z, - [z,,i Rdym Ji=1 and decode them as (Z., E dR Ji=l using m parallel multilayer perceptrons. We then recombine the elements with the same position in vectors {2,,) into dr vectors {ur, E 18")) and further decode them as {ūr.j E di 147 parallel multilayer perceptrons. Finally, we obtain the causal matrix W, ddmxdr;
We introduce a purallel decoding strategy to reduce parameters and computations (Fig. 2
Jj=1 with dr
[ ... .../ūr ür, MLP)(o(uu.))
(4) where (-) denotes the multilayer perceptron and a(-) ) is an activation function. Compared with direct decoding, our strategy reduces the number of parameters from (d2m² to (d' + d and the computational complexity from O O
fugl E [1, dr])
recombine
(ž,, [1,m]] ž,, MLP.(((((..)
The causal matrices obtained above are separated by time steps and thus remain in coarse- coarse-grained grained dynamics. We approximate their fine-grained piecewise linear trajectory by supplementing the
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causal matrix at each time step with linear interpolation. To this end, we further assume that S is small enough based on Eq. (2). and the variation of the causal matrix in the interval [t, t S] can be modeled deterministic linear function by the first-order Taylor formula:
W *++ W; (W++S Wi). s, SE [0 S
(5)
The above formula is equivalent to linear interpolation, and we estimate the causal matrix at each time step, while only calculating |(T - K)/S] matrices in Eq. 44p. Although smooth aar-varying trajectory of causal structures requires a small S, we can easily cut the computation by more than half with S 2 2.
4.2 MODIFICATION OF THE ACYCLIC CONSTRAINT
The goal of an acyclic constraint to prevent variables from self-reconstruction to identify reliable causal structures. Existing acyclic constraints (heap Tr(e Tr(eW.W) - dd (Zheng et al. 2018 and (hpoty Tr((1 1/ W W) d) YYu et al. 2019) rely the trace of the matrix measure the cyclic characteristics. Subsequent studres (Zuu et al. 2020 Lachapelle et al. 2020 ZZheng et al. 2020b, Kyono al. 2020) basically follow these two constraints, learning causal structures with the augmented Lagrangian format and the L-BFGS-B algorithm (Nocedal & Wright 2006), and result unacceptable inefficiency. Although the log-det acyclic constraint (hiog = log (al
W o W) + d log a)) (Bello et al. 2022) improves efficiency, it remains in finite optimizable space W = (W> E Rdxd pp( W W) < a. Once p( W o W) 2 a, the model is forced to reduce the learning rate and retrain, this severely undermines the efficiency and stability of causal discovery.
well known that the norm of matrix is larger than (or equal to) its spectral radius. Thus, suggest scaling W o W by its 1-norm to keep always differentiable and optimizable, giving the following new acyclic constraint:
We propose an efficient and effective log-det acyclic constraint.
Wo W
hormm = - log det(al
+ dloga a
(6)
| o wili
In this constraint, a is a constant(close to 1 but strictly larger, e.g. 1.001, defined as 1+) and does not need additional adjustments, and I|W Wili is the 1-norm of W o W. The scaled matrix always within the optimizable space (p( W[Wo < 1+) and avoids redundant retraining and
hyperparameter search. Although our constraint penalizes the cyclicity of W-W
[Wo the cyclic
characteristic of WoW is consistent with that of W W, and WeW is DAG if and only [[o
TWoWi In
2 if W o W is DAG. In addition, Vhnorm = || (al ||Wow|| --T o W and Vhtog
WoW
2(al Maal-Woo Wow) oW also have the same characteristic, and Vh if and only if Vhttg = 0.
woW Theorem 1. Characterization. Given the acyclic constraint haorm = log det(aT- wwolll/ ) doog a. Then, the following holds:
(i) hnorm 2 O, with hnorm 0 if and only W DAGs.
(ii) Vhnorm l|woll|2 (al DAGs.
WoW
W, with Ahnorm 0 if and only if W
||WoW]]
It follows from Theorem I that hnorm inherits the excellent characteristics of htog. Referring to Definition 3.3 in Nazaret et al. (2024), our acyclic constraint satisfies three stability criteria:
Theorem 2. Stability. The acyclic constraint hnorm is stable. For almost every A WoW Rdxd:
[[Wol]]
(i) E-stable hnorm (kA) Ok-+to0
(ii) V-stable hnorm (A) + Knorm (kA) Pa-so+ (1).
(iii) D-stable hnorm and Vhnorm are defined almost everywhere.
E-stable and V-stable ensure that hnorm neither explodes to infinity nor rapidly vanishes to 0. stable means that the acyclic constraint and its gradient always computable for almost every A.
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Algorithm DyCausal: Coarse-to-fine learning of dynamic causal structures
Input: Cause series X effect series V. penalty coefficient 9, decay coefficient (default is 1), decay rate 7 (default is 0.1), number of iterations r (default 4) and threshold 8
Output: Dynamic causal structures G {Gr]=r+1
1: Initialize parameters 0
2: for epoch = 1 to r do
3: Reconstruct effect series: Yo.+K W.X
4: Minimize the objective: min /4( NTK Eierti Y - + + hnorm
5: Update coefficient μ.: 1a = 7/14
6: end for
7: Prune W (W, 17 : 9 (W; > 8)
8: return G
In particular, (p(A) may not be calculable A has repeated eigenvalues or if ((A) corresponds to multiple eigenvalues. Therefore, hnorm constrain the cycle more D-stably than directly using the spectral radius (Nazaret et al. 2024). We prove Theorem in Appendix
4.3 RECONSTRUCTION AND OPTIMIZATION
We now define score-based optimization objectives (Sun et al., 2023 Cheng et al. 2025) to train the model to identify the causal structures. We rearrange the observed time series, and build the cause series X = [X; == .x..... XX [x and the effect series ) = (Y, = X,)
The score-based objective claims that a causal model consistent with the data distribution should accurately reconstruct the effect series from the cause series and satisfy the acyclic constraint. Given
the fine-grained causal matrices W = (W, E Rdmxdr]T ) estimated Section 4.1 we set
1 to represent the linear causality and define the reconstructed effect series Y Yit+K
W, W,Xi 1-tKK. Minimizing the reconstruction loss with the acyelic constraint Section
leads to
the optimization problem:
T-K K
arg min W.
Σ
YY - Y Yatk + Weh
NTK
subject hnorm (Wi L..,,
where 0 is the parameter of the convolutional network and 33 is a penalty coefficient. The observa- tions in [t, + K] are reconstructed by the same matrix W., which is equivalent to adding samples learn stable W, each time step. We include the I regularizer to promote sparse matrices and adopt the central path approach to solve this optimization.
The DyCausal procedure summarized in Algorithm I We multiply the minimization objective by coefficient and then add the acyclic constraint as a regularization to constitute new objec- tive. In each iteration, we reduce μ and minimize this objective. After the optimization, we prune the small elements of the causal matrices {W.) W with threshold ö, and obtain the dynamic causal structures G. We discuss the identifiability of DyCausal Appendix D
Extension to the nonlinear model. By identifying the fully dynamic causal matrix, DyCausal can flexibly adapt to nonlinear causal models through parameter adjustment. Inspired by (Zheng et al. (020b), which replaces the partial derivatives of the reconstruction model with the parameters of the first-layer network to capture nonlinear causality, we build a reconstruction model and treat to represent the number of hidden units and
the causal matrix W, as its first layer. We set introduce additional multilayer perceptrons to reconstruct the effect series as follows:
YL+KK- MMCP((
(8)
where o(-) is activation function, MLP {MLP]] is composed of independent mul- tilayer perceptrons with input size m and output size 1. We keep the optimization process in Al Algorithm [ except for the additional optimization of parameters in MLP. By reshaping the

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causal matrix W, from Rdmxdr to Rdxmxdr get the new weighted adjaceney causal matrix W, V Σ (Wel..::)? W. E Raxdr,
Extension to the ordinary differential equation model. DyCausal can also be extended to the currently popular ordinary differential equation (ODE) (Chen et al. 2018 De Brouwer et al. 20200] Bellot et al. 2022), which widely used in the field of biology Zhu et al., 2012; Zheng et al,
2020a Wang et al. 2023). The biggest difference between ODE and SEM is that ODE describes the generation process of change rates rather than changed variable values. We adjust the reconstruction process based on the nonlinear reconstruction model follows:
Y Ya+kK = YY-1s-14KK MCCP(
The reconstruction process of ODE does not involve instantaneous causality. so there is no need to penalize the cycle in the causal matrix, which implemented in Algorithm [ by removing hmorm and setting the number of iterations to r 1.
EXPERIMENTS
We conduct extensive experiments on synthetic and real datasets to evaluate the effectiveness of DyCausal. To compare the performance of DyCausal with all baselines, we follow existing work (Cheng et al. 2025 Zheng et al. 2018) and report three widely used metrics: TPR (true positive rate), SHD (the difference between real and estimated graphs, including the number of reversed, missing, and redundant edges), and F1 (the harmonic mean of precision and recall).
Synthetic data. We use the ER model (Erd6s & Rényi 1960) to generate random ground-trute causal graphs. We then use the additive noise model and the Lorenz model (Lorenz) 1996) to simu- late the structural equation and the ordinary differential equation, respectively, and generate the time series. A more detailed data generation process is given in Appendix B.I
Real-world data. To demonstrate the broad applicability of DyCausal to real-world problems, apply to four real-world datasets: CausalTime (Cheng et al.) 2024b) records time series in three real scenarios, including weather, traffic, and medical; NetSim (Smith al. [01IT) records the time series of glood-oxygen-leve dependent signals in different brain regions: DREAM-3 (Prill et al., 2010) and Phoenix (Hossain et al., 2024) record the time series of gene expression levels.
Hyperparameter settings. We conduct experiments using the public code available for the com- parison baselines. Detailed hyperparameter settings for our DyCausal and baselines are stated in Appendix B.2, which are sufficient to play the performance of the baselines. We also perform hyperparameter analysis in Appendix C.I1 which proves that we have selected the optimal hyper- parameter for DyCausal.
5.1 RESULTS ON SYNTHETIC DATA
We first compare DyCausal 1 against state-of-the-ar methods to demonstrate its performance in dynamic temporal causal discovery, including DyCAST (Cheng et af.) 2025) for partial dynamic temporal causal discovery, DYNO (short for DYNOTEARS) (Pamfil et al. 2020), NTS-NO (short for NTS-NOTEARS) (Sun et al. 2023) for static temporal causal discovery. Following Appendix B.I we generate ground-truth causal graphs and then assign weights each edge. To simulate dynamically changing causal structures, we let the weights of the edges in the causal graphs be the sine or cosine functions. We simulate two dynamic scenarios: one where only instantaneous causality changes over time, and the other where all causality changes over time. We generate N 200 time series with length T = 10 and maximum lag 1. F shows the evaluation of the causal structures identified by DyCausal and by each baseline at each time step.
clear that DyCausal outperforms other baselines and identifies the most accurate dynamic causal structures at all time steps. DYNO and NTS-NO have decreased performance at both ends of the time series. This is because they are insensitive to the changing data distribution, confuse the distribution of observations both ends and in the middle of the series, and fail identify dynamic causal structures. DyCAST is designed for dynamic instantancous causality and thus robustly iden- tifies causal structures each time step in the corresponding scenario, but with less accuracy than

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HyCona
BYAD
STE.N
IVCAST
FDR
M.
(a) Results on time series with dynamic instantancous causality.
HyConel
1950
N72000
SHD
14 M1
19,
R. Me W. W. Mr W. Wio
(h) Results time series with fully dynamic causality.
Figure 3: Results on time series with dynamic causality. (WI,-.. W represent the causal graphs estimated at each time step.
Table I: Results on time series with dynamic causality. W and W50 represent the causal graphs estimated at both ends of the time series. W25 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.
nodes
80 nodes
t
Methods
TPR1
FIf
SHD
TPR
FIf
SHD
DyCausal
6.954
91.43++5.60
7.40
89.10=1.70
93.85=1.31
38.20
DYCAST
25.98=10.33 10.61=10.67
45.43:17.21
W1
DYNO
3.59+3.18 3.524:2.90
18.40:10.75
0.35=0.70
0.34
294.90+28.51
NTS-NO
3.89+1.57
4.03+11.69
12.04
0.77+1.00
0.78+1.02
294.20+15.55
DyCausal
90.9316.01
94.3813.90
4.10-2288
91.01=1.97
95.20=1.118
13.40:2.15
DYCAST
50.88=11.29 11.12
23.00-1..63
W25
DYNO
7..64+6.77 69.20+
25.00+6.91
69.49+4.70
72.81
82.90+17.22
NTS-NO
30.99+5.224
33.00+7.37
10.56+
32.36-4.23
33.28+4.07
10+10311
DyCausal 87.2916.60
90,3815,99
8.20
87.55
2.89
1.78
22.00 4.600
DYCAST
24.15:18.97
28.64:19.87
44.1411..75
W0
DYNO
3.59+3.11
3.52+2.90
10.40
0.35+0.70
0.34+0.68
294.90+-8.551
NTS-NO
3.89+1.57
4.03+1169
72.03+
0.77+1001
0.78+1.02
294.20
DyCausal. Limited by the ordinary differential equation to model the time-varying trajectory of causality, DyCAST struggles to identify causal structures with drastically changing weights. Only DyCausal succeeds in time series with fully dynamic causality, thanks to coarse-to-fine modeling of time-varying trajectory with instantancous and lagged causality.
Performance time series with dynamic causal structures. We run DyCausal and baselines on longer time series with larger causal graphs to further demonstrate the superior performance set the length of the time series as T 50 and the number of
of DyCausal. Specifically, nodes in the causal graphs at = {20, 80). Table [ presents the evaluation of the causal graphs identified by DyCausal and the baselines three key time steps
{1, 25, 25,5 We can that DyCausa 1. is significantly superior to all baselines and its advantage is especially prominent both ends of the series (t and t 50). Compared the static methods DYNO and NTS-NO, DYCAST identifies partially dynamic causal graphs but fails to learn causal matrices with opposite weights both ends of the series. This indicates that the ordinary differential equation modeling the vire-ngrying trajectory of the causal matrix cannot identify the weights with large variations. In addition, DYCAST models the causal graph at each time step and thus fails to converge on graphs with 80 nodes. In contrast, DyCausal only encodes causal graphs for partial time steps and then refines them to each time step by linear interpolation, improving stability and accuracy. Appendix C.13 elaborately presents the evaluation of the instantaneous and lagged causal graphs identified by DyCausal 1 at each time step, further demonstrating the superior performance DyCausal in identifying fully dynamic causal relationships.
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1
Figure 4: Comparison of acyclic constraints and their gradients, and runtime.
We further provide ablation experiments in Appendix C.2 prove that linear interpolation in DyCausal is necessary for identifying dynamic causality. We also evaluate DyCausal on time se- ries with non-smooth dynamic causality. The results in Appendix(C.I2Jndicaad that DyCausal can still effectively identify causal graphs with considerable accuracy. We run DyCausal on datasets with more variables, fewer samples, longer lag and different series length and noise strength in Appendices C.3 and the results show that DyCausal 1 is scalable and robust.
Performance on time series with static causal structures. We conduct comparative experiments to show that DyCausal is also capable of identifying static causal structures from time series. In particular, we synthesize time series with three models: the linear and nonlinear structure equation models (SEM) and the ordinary differential equation (ODE). Comparison baselines include SVAM (Hyvärinen et al. 2008), PCMCI (Runge et al. 2019), and TECDI (Li et al., 2023) for all models, DYNO for linear SEM, NTS-NO for nonlinear SEM and NGM (Bellot al. 2022) for ODE. We generate ground-truth causal graphs and time series with length T 1000 and maximum lag 7 = as described in Appendix B.I To adapt to algorithms that require multiple time series, we divide the original series into N 20 time series with length 7 = 50, which ensures consistent sample sizes and fair comparisons. We report the results of DyCausal and all baselines in Appendix C.9 The results show that DyCausal consistently improves accuracy under three models and attains a sig- nificant speed-up. We also compare the acyelic constraint horm against with the original constraint htgg in Appendix C.10, and prove that the improvement of hnorm. is effective and necessary.
5.2 EVALUATION OF THE ACYCLIC CONSTRAINT
We have proved in Theorem 2 and Appendix A that hnorm meets the stability criteria. We provide empirical results to further demonstrate the superiority of hnorm over existing acyclic constraints, including heap (Zheng et al. 2018), hpplly (Yu et al. (2019), bigg (Bello al., 2022). and ho (Nazaret et al., 2024)). We compare these acyclic constrants in two types of matrices: uniform random matrices k), and cycle graphs with nodes where each edge weight is 0.5 or -0.5. Figure shows the curves of the acyclic constraints and their gradients, and runtime varying with k and d.
As l and d increases, Marp and hyply rapidly grow to infinity or vanish to 0, reflecting that they are E-unstable and V-unstable. htog performs robustly on uniform random matrices but its performance decreases rapidly with increasing on eyelic graphs. This because small weights reduce the sensitivity of hlog to long cycles. In contrast, hnorm robust against the rapid expansion and attenuation, and remains stable and efficient in all experimental settings. Although h2 is similarly stable both types of matrices, it has very small gradient and exponentially increasing runtime on cyclic graphs. We observe that cycle graphs have close eigenvalues, which could be the reason for the disappearance of Vhp- A small Vh, means that h, is prone to getting stuck in local optimal solutions and cannot effectively punish the cycle. Therefore, the acyclic constraint hnorm is the most stable and efficient, and also the most effective in punishing cycles.
5.3 RESULTS ON A REAL CASE
We evaluate DyCausal and other baselines on CausalTime (Cheng et al. 2024b). In particular, we introduce an additional baseline, CUTS+ (Cheng al. 2024a). which targets at high-dimensional and sparse real-world data, and JRNGC (Zhou et al. 2024). which identify causal structures with the Jacobian matrix. Table shows detailed comparison of the results.
We observed that DyCausal achieves the best Precision and F1. Higher precision means that DyCausal 1 can identify more reliable and trustworthy causal structures (with most of the edges cor-

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Published as a conference paper at ICLR 2026
Table 2: 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 Fl simplify the representation.
Traftic
AQI
Medical
Methods
Precision
F1f
Precisiont
FIf
Precisiont
F1
DyCausal
87.4111.885
54.11:2.51
75.31=1.43
13.16::1223
83.21=2.75
1.38
CUTS+
80.10+3.16
1.72
65.60-3.47
58.20+2444
44.82+3.55
30.23++1.85
JRNGC
68.04455.56
52.0442.41
72.34+3.43
55.36+1.61
55.56--0.63
52.05++0.75
NTS-NO
67.43:15.35
4..07=1.85
72.70=2.99
40.0911.29
44.80=2.13
34,27=1.89
DYNO
47.631:3.26
45,34:12.92
1.355=1.34
44.60110.75
48.40=2.07
31.70.:..25
SVAM
45.45++1.39
4..26+1.14
55.22-0081
50.26++0.78
49.33+0041
41.63++0.2
PCMCI
22.61+2.98
25.30+3322
31.74+4.51
39.26+7.772
32.71+5.21
37.39+5.46
NGM
21.2112.15 30.7611..17
28.38
31.76.12.16
37.54
14.15
TECDI
70.05
1..01
2.81
47.44+2.35
44.22++3.58
36.51
315
1800
-664
Time
Time
Timee
Time
Figure 5: Estimated dynamic causal structures on traffic subset. We observed that: (i) causality I6 + 24 24 + 219, 26 + 219, 31s + 27 and F19 - Is are dynamic up to the time 20 and + Ig, #19 J4 and Zie - 2 are static before and after t= 20, but
static afier:(ii) causality with different causal strengths. particular, the strength of x4 + Jg is almost zero before 20, which means that 24 28 only be identified from time series after t 20.
rect) when facing datasets without verifiable causal graphs. CUTS+ identifies the summary causality without limiting the window length and is inferior to DyCausal. This indicates that the window causality is closer to the true causality CausalTime than summary causality. Figure visualizes the dynamic causal structure identified by DyCausal in the traffic subset, which records the flow changes of 20 traffic nodes within a day. It can be seen that the dynamic causality is clearly divided into two change patterns by time step changes in causal relationships that other algorithms often treat as noise, preserving true structures while avoiding the inclusion of redundant ones. Appendix CC.44 provides the results of DyCausal in other real datasets, NetSim (Smith et al. 2011), DREAM-3 (Prill et al.) 2010)) and Phoenix (Hos- sain al. 2024), where DyCausal identifies the most accurate causal graphs.
20. DyCaus al accurately capture the dynamic
CONCLUSION
In this paper, propose DyCausal, novel framework to overcome inefficiency and instability in fully dynamic temporal causal discovery. Compared to previous methods, DyCausal learns the time-varying trajectory of causality from coarse to fine with the most advanced performance. The theoretically and empirically stable acyclicity constraint, enforced via the matrix norm, further enhances both the robustness and efficiency of DyCausal. 1. Extensive experiments demonstrate that DyCausal consistently performs well in dynamic, partially dynamic, and static time series, adapts well to real-world datasets, and is sensitive to the fluctuation and evolution of causality.
This work assumes that the time series is regular. Although previous work (Gong et al. 2015 [seki et al. 2019) attempted to solve irregular time series, this is not the focus of this work. Dy a sal the existing completion strategies (Cao et al. 2018) Cini al. 2022, Jin et al. impute irregular time series and then identify dynamic causal structures. In the future, we plan to expand DyCasual to causal system where variables dynamically participate leave.
2024 to
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7 ACKNOWLEDGMENTS
This work is partially supported by National Key Research and Development Program of China (No. 2024YFF1206604), National Natural Science Foundation of China (62531013, 62272276 and 62432006), Shandong Provincial Natural Science Foundation (No. ZR2024JQ001 and ZR2025LZH004) and Taishan Scholars Program (No. tsqn202408317).
ETHICS STATEMENT
We promise that we have read the ICLR Code of Ethics, and this article has not raised any questions regarding the Code of Ethics.
REPRODUCIBILITY STATEMENT
We promise that DyCausal is reproducibility. We provide the code of DyCausal at including the code to run, synthesize datasets,
http://www.sduu-idea wttp.///ooodes.ppnaameeeeeee and perform experiments. The code also includes the real datasets used in the experiments.
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A PROOF OF STABILITY FOR THE ACYCLIC CONSTRAINT
We recall Theorem 2 and prove it.
Theorem 2. Stability. The acyclic constraint hnorm is stable. For almost every A E Rdxdd
(i) E-stable hnorm (k:A) (0x-0001).
(ii) V-stable fnnrr ()) o = knorm (ko) = Qk- (1).
(iii) D-stable Nnorr and V Vhnorm are defined almost everywhere.
Proof. Since the 1-norm is the maximum sum of the absolute values of the elements in each column of the matrix, if the matrix A is multiplied by the constant its 1-norm is also multiplied by the constant k. Therefore, we have the following conclusions:
E-stable. aorany k: >> and matrix W, -nnom (k:A) = log det((ll
+dloga [A
AA
A. log det - [ + log a = Knorm (A) A)= = +o (1).
V-stable. For any > 0 and matrix such that hnorm (A) > hnorm (kA) - log det | + d
log det(al (Tlli + d dloga a knorm (A)
Px-+0+ (1).
According to Theorem hnarm inherits the excellent characteristics of hing and Whnorm is well defined within the optimizable space. Furthermore, the spectral radius of the matrix scaled by the A 1-norm is always within the optimizable space (pl T < 1+). Therefore, bnorm and its gradient
Vhnorm well defined everywhere, and haorm is D-stable.
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B EXPERIMENTAL DETAILS
B.1 DATA GENERATION
In this section, we provide a detailed process for generating the simulation data in experiments.
Ground-truth graph. Given the number of variables d, the maximum lagged length and edge density e, we randomly generate ground-truft -ruth graphs. Specifically, we use the ER model (Erdós & Rényi 1960) to generate a DAG with d nodes and ex edges as the instantaneous causal graph G We then set r zero matrices in Rdxd each zero matrix, the elements are set to with probability e/d and the lagged causal graph Glag is obtained. We vertically concatenate the instantaneous and lagged causal graphs to obtain the complete temporal causal graph G. In our experiments, we always set the edge density e =
Linear structural equation model. We use the linear additive noise model to simulate the linear structural equation generate data as follows:
X, = - ....--.......... [x-,]
(A1)
where C sampled from the Gaussian distribution N 1). To assign weights to each nonzero ele- ment in the ground-truth causal graph, uniformly sample weights from the interval [-0.5, 0..3 [0.3, 0.5] to obtain a causal weight matrix W. Furthermore, we multiply each weight from time to [ r by decay parameter (1 (1/n)², where
and [0,-]: This strategy is widely used by existing methods (Pamfil et al., 2020; Cheng et al., 2025) to reduce the influence of variables that are further back in time relative to current time. In experiments, always 17 1.5.
Nonlinear structural equation model. We use the nonlinear additive model parameterized by multilayer perceptrons simulate the nonlinear structural equation to generate data follows:
Ft (MP( +
(A2)
where e is sampled from the Gaussian distribution (0, 1). o()) is a sigmoid function and PA(x;,))
parents of Ft in the interval it 2-77 -, t). We set multilayer perceptrons with 1 hidden layer and 100 hidden units and uniformly sample perceptron parameters from the interval [-2, 0.55 U [0.5, 2].
Ordinary differential equation model. We use the Lorenz model (Lorenz 1996). widely used ordinary differential model, to generate data as follows:
/ -1,5--
21,441) Fz,i F + dw
(A3)
where F is constant and du is sampled from the Gaussian distribution N(0, 0.05). We randomly sample data from the Gaussian distribution NO.,0.00 as observed data at time = O, and then gradually generate data at time > 0 according to the above model Z441,1 It, dry, In our
experiments, we always set F = 5.
Dynamic model. Given the causal weight matrix Wo. we set its weights to be time-indexed sine or cosine function to simulate the dynamic causality. Specifically, we uniformly sample a probability matrix Wp p E d and define the dynamic matrix W, follows:
Woli.j.oss (TTOOs(TTT/)////////))(( t) if HWW 0.5 Woli, (in((i/(((TTT)
W. .l =
(A4)
wf [4. > 0.5
where is the length of the series. Given the above weight matrix, the observed data at time t is only generated by W..
B.2 PARAMETER SETTING
In this section, we provide the detailed parameter settings of all algorithms used in experiments. For fair comparison, we search for parameters such as regularization coefficient, learning rate, and prun- ing threshold that do not affect the model size to achieve the best performance of the baselines. In particular, we set DyCausal, NGM and NOTERTTSS to have the same reconstruction model size to fairly compare their performance. All experiments were 18.04.5, Intel Xeon Gold 6248R and Nvidia RTX 3090). The code of DyCausal all is shared at http://wwww....deeee.emcc/eeeepp
the same server (Ubuntu
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DyCausal
The window size K 10 for static and K = 2 for dynamic
The sliding step size 5 5 for static and $ 4 for dynamic
The model contains hidden layer and 10 hidden units for nonlinear data
The learning rate = 0.005
The sparse term coefficient / = 0.05
The pruning threshold 8 = 0.3
PCMCI
-The conditional independence test uses the Pearson correlation coefficient Significance P value PC. 0.05
SVAM
'he pruning threshold = 0.3
NGM
The Lasso regularization coefficient 0.05
The model contains hidden layer and 10 hidden units
The batch size batch_size = 20
The predicted length horizon 5
-Te pruning threshold 8 0.1
DYNOTEARS
-The sparse term coefficient / = 0.1
-The pruning threshold 8 0.1
NTSSNNTEARS
-The sparse term coefficient / = 0.01
The model contains hidden layer and 10 hidden units
The pruning threshold 8 0.5
TECDI
The model contains hidden layer, and each layer contains 16 hidden units
The learning rate Ir 0.005
The Regularization coefficient 0.3
DYCAST
The model contains hidden layer, and each layer contains 16 hidden units
The learning rate Ir 0.001
The hidden units is 0.8d
The pruning threshold 0.1
C ADDITIONAL EXPERIMENTS
C.I RESULTS ON TIME SERIES WITH DYNAMIC CAUSAL STRUCTURES
We demonstrate the performance of DyCausal on time series with dynamic causality. We follow Appendix B.I to generate ground-truth causal graphs and then assign time-varying weights to each edge. We let the weights of the edges in the causal graphs be the sine or cosine functions of the time index, thereby simulating the dynamic causal relationships. We generate N 200 time series with length 7 = 50 and maximum lag t 1. Table Al shows the results of DyCausal and the baselines at three key time points (5,
We can see that DyCausal is significantly superior to all baselines and its advantage especially prominent both ends of the series (t and 50). Compared to the static methods DYNO and NTS-NO. DYCAST identifies partially dynamic causal graphs but fails to lear causal matrices with opposite weights at both ends of the series. This indicates that the ordinary differential equation
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Table Al: Results on time series with dynamic causality. W and W50 represent the causal graphs estimated at both ends of the time series. W25 represents the causal graph estimated in the middle of the time series. We omit the percentage signs % of TPR and F1 simplify the representation.
20 nodes
80 nodes
Methods
TPR1
FIf 91.43:1..60
SHD
TPR
FIf
SHD
DyCausal
6.95
7.40:1:5.39
89.10
1.31
18.20
DYCAST
1198=100.33
10.61
45.43:17.21
DYNO
3.591:.18
3.52+42.90
18.40
0.35=0.70
0.34
294.90=28.51
NTS-NO
3.89+1.57
4.03:11.69
72.33+12.04
0.77=1011
0.78
294.20=15.55
DyCausal
6.00
94.38+13.00
4.10-+2.88
1.97
95.20
13.40
DYCAST
50.88=11.29 11,12
23.00
W25
DYNO
71.64:6.77 69.2011.996
25.00116.91
69.49-4.70
72.81
82.90
NTS-NO
30.99115.24 33.0017.37
10.7610..74
32.36-4.22
33.28:14.07
196.10116.31
DyCausal
87.2916.60
90.38.15.99
8.20
87.55 2.89
92.82-1.778
4.66
DYCAST
24.15:18.97
28.6411987
44.14116.75
50
DYNO
3.5913.18 3.52-12.90
48.40
0.35
0.34
294.90
NTS-NO
3.89
1.03+1166
72.33+12.04
0.77
0.78+1.02
294.20
W
W2, 25
Ws
W
W 25
W 50
1.0
0.5
-0.5
-1.0
-1.5
-2.0
Ground-truth
Estimated
Figure A1: Results on dynamic settings with 10 variables and maximum lag. W1 and W represent the causal graphs estimated at both ends of the time series. Wa 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 the weight. The estimated matrices are basically consistent with the ground-truth matrices.
modeling the time-varying trajectory the causal matrix cannot identify the weights with large variations. In addition, DYCAST models the causal graph each time step and thus fails to converge on graphs with 80 nodes. In contrast, myCausal only encodes causal graphs for partial time steps and then refines them each time step by linear interpolation, improving stability and accuracy. We further compare the causal matrices estimated by DyCausal and the ground-truth causal matrices in Figure where DyCausal not only identifies the causal graphs, but also restores the weights of the causal matrices.
C.2 ABLATION FOR LINEAR INTERPOLATION IN DYNAMIC SETTINGS
Linear interpolation necessary for large-scale causal graphs and long time series. It enables the coarse-grained causal matrices learned by the window to jointly participate in the training; oth- erwise, each matrix can only be trained in isolation based on the data within the corresponding window. We introduce ablation baseline DyCausæl w/o inter in dynamic settings, which only allows the causal matrices encoded by the sliding convolutional window to participate in the series reconstruction without approximating the fine-gh finengrined time-varying varying trajectory of the causal matrix through linear interpolation. Table A2 shows the performance comparison between DyCasual and DyCausal 1 w/o inter.
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Table A2: Results on time series with dynamic causality. W and W50 represent the causal graphs estimated at both ends of the time series. W25 represents the causal graph estimated in the middle of the time series. We omit the percentage signs % of TPR and F1 simplify the representation.
10 nodes
20 nodes
Methods
TPR
FIf
SHD,
TPR
Flf
SHD
DyCausal
90.35:16.62
91.42.14.39
3.40
89.86:13.83
92.96
6.00:12.37
W1
w/o inter
75.67-122220
72.98+:23.78
12.50+13.86
90.06-15,45
92.18
6.10::3.67
DyCausal
90.41
92.65+4..73
3.10
91.09::5.06
93.64:15.47
4.90:1:.557
W
w/o inter
15.66=+1..05
83.25++20.30
1.50==11.15
18.09
91.28-+9.88
6.40+7.30
DyCausal
69.62:1..9
90.8113.67
3,50=1433
87.7516.53
89.4011.84
8.3014.00
30
w/o inter
16.26 72.83+20.28
12.80+13.31
12.45-12.35
17.22
9.50
Table A3: Results for settings K 4,5
and K
1,5
DyCausal
K
4,
4,5=1
=1
K
1,
=
TPR
F1
SHD
TPR
F1
SHD
TPR
F1
SHD
W1 1
91.56
92.73
63.0
90.54
90.95
72.0
83.33
20.48
2485.0
W 25
96.70
98.32
13.0
95.47
97.68
18.0
89.22
30.22
1644.0
W 50
94.18
94.18
46.0
90.09
92.59
64.0
85.68
20.76
2512.0
runtime (S)
2926.0
3527.0
1970.0
The results show that DyCausal w/o inter achieves worse performance than DyCasual. This shows that linear interpolation in DyCausal is not only a core component for approximating fine- grained dynamic causality, but also plays an important role in identifying dynamic causal discovery. Linear interpolation links the isolated causal matrices inferred in each sliding step, enabling data that are originally not within the sliding window to participate in the reconstruction process and guide the learning of the dynamic causal matrix. Therefore, DyCausal achieves improved precision and stability in dynamic causal discovery. On N 200 time series with length T = 50 and d 200 nodes, we further estimate the causal graph at each time step under two settings. One sets the window size K and the sliding step S = The other sets both the window size and the sliding step to one. The results are reported in Table A3
The results show that DyCausal runs for a longer time in the former setting. That is because the model encodes more matrices and calculates the acyclic constraint term more times. DyCausal achieves reduced results in the latter setting. This is because the sample size within a window of size 1 is too small support the model in learning causal graphs with 200 nodes.
C.3 SCALABILITY TO CAUSAL GRAPHS WITH MORE NODES
We run DyCausal on ground-truth causal graphs with more nodes to demonstrate scalability for large-scale data. We continue to use the dynamic settings in Section C. and expand the number of nodes to d = (50, 100, 200, 3003. Figure A2 shows the results of DyCausal at three key time points t {1, 25, 50).
We can see that DyCausal maintains excellent performance on more nodes with TPR greater than 0.78 and Fl greater than 0.82, which proves the scalability of DyCausal in handling large-scale causal graphs. It is important to note that with increasing number of nodes, we reduce the learning rate to accurately learn causal structures. The combinations of the number of nodes and the learning rate are (50, 5e - 5). (100, le - 5), (200, 3e - 6), (300, 1.5e 6). This may be because a smaller learning rate can promote the convergence of the model to identify accurate causal structures.
C.4 ROBUSTNESS TO FEWER SAMPLES
We run DyCausal on datasets with fewer samples to demonstrate the robustness of DyCausal to sample size. We also follow the dynamic settings in Section C.I and adjust the sample size N = (20, 50, 100, 150). Figure shows the results of DyCausal on graphs with 50 nodes three key time points L 25,
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155
632
-15
Runtimee
3.09
SIID
200
0.5
Namber modex
Namber modes
Namber aodes
Number of modes
Figure A2: Results of DyCausal on datasets with number of nodes d = (50, 100, 200, 300).
62
90
OeCraa
300
248
Runttiors
390
Sample
Sample
Sumple
Sample
Figure A3: Results of DyCausal datasets with fewer samples N 1220, 50, 100, 150).
We can see that DyCausal achieves a TPR of greater than 0.5 with only 20 samples and maintains similar performance with 50 samples as it does with 100 and 150 samples. Surprisingly, the runtime of DyCausal for 150 samples is less than that for 50 and 100 samples. This may be related to the memory scheduling of the GPU. Despite this, DyCausal still provides an acceptable runtime (less than 11 minutes). All of the above indicate that DyCausal has excellent robustness with fewer samples.
C.5 ROBUSTNESS TO DIFFERENT TIMES SERIES LENGTHS
Still under the dynamic settings in Section CT1 we change the time series length verify the robust- ness of DyCausal 1 to different series lengths. Figure shows the results of DyCasual on datasets with time series lengths T {10, 30, 50, 90).
As the length of the time series increases. DyCausal achieves gradually improved and stable per- formance. In shorter time series, the data distribution changes more drastically within the slid- ing window, resulting in decreased performance. However, DyCausa 1 still achieves TPR and F1 around 0.7 with
10. The runtime DyCausal T = 70, 90 significantly longer than at T 10, 30, 50. One possible reason is that longer time series occupy more memory, leading to more memory swap operations.
C.6 ROBUSTNESS TO DIFFERENT NOISE STRENGTHS
We further verify the influence of noise strength on DyCausal, In the dynamic settings in Section set the noise strength (0.5, 1.0, 2.5, 5.0}. and report the results of DyCausal under these settings in Figure A5
C.I
Larger noise, except for slightly increasing the runtime of DyCausal, does not significantly affect the performance of DyCausal. In contrast, DyCausal achieves a decrease TPR and F1 with noise strength 0.5. This may be because smaller noise makes it difficult for the model to distin- guish between direct and indirect causal effects, leading to the identification inaccurate causal structures. The above results prove that DyCausal perform stably under different noise strengths.
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DyCrani
Runtiness
SHD.
0.75
0.70
16
length
length
length
Figure A4: Results of DyCausal datasets with times series lengths {10, 30, 50, 70, 90)
W1
W0
0.9
380
Runtfancss
0.8
Nodse
Noise
Figure A5: Results of DyCausal on datasets with noise strength {0.5, 1.0, 2.5,5.).
C.7 ROBUSTNESS LONGER LAGS
We run DyCausal on datasets with longer maximum lags verify its robustness against differ- ent lags. the dynamic settings in Section C.I. we set the number variables 50 and the maximum lags 2,, 4, 5), and report the results of DyCausal in Figure AA6
We can observe that DyCausal maintains stable performance under longer lag. When the maxi- mum lag 5, DyCausal achieves true positive rate of around 0.7 and an Fl of around 0.75.
C.8 ROBUSTNESS TO UNKNOWN LAGS
We further explore the robustness of DyCausal to unknown lags. In the same dynamic settings, we set the maximum lags t 3 to generate time series. We then let DyCausal learn temporal causal structures with estimated maximum lags r (1, 3, 5). For the case with < 3, we fill the learned causal graphs with 0, and for the case with >> 3, we only retain the structures of the graphs where the lag is less than or equal to 3. Figure A7 shows the results of DyCausal.
We can clearly see that as the estimated lag increases, DyCausal achieves higher true positive rate. Although an overly larger estimated lag also reduces performance, we suggest choosing a slightly larger estimated lag for DyCausal when the maximum lag is unknown.
C.9 RESULTS ON TIME SERIES WITH STATIC CAUSAL STRUCTURES
We generate ground-trute -ruth causal graphs and time series with length T 1000 and maximum lag == 2 as described in Appendix B. To adapt to algorithms that require multiple time series, we divide the original series into N 20 time series, each with length consistent sample size and a fair comparison. Figure A8 shows the results in the settings with linear and nonlinear structural equation models and the Lorenz model.
50, which ensures
We note that DyCausal attains significant speedups against TECDI, NGM, and NTS-NO (espe- cially for the case with S0 nodes), which attributed to our improved acyelic constraint. What more exciting is that DyCausal consistently achieves improved structural precision (low- est SHD) and competitive TPR and Fl under three causal models. This could be attributed to the fact that the reconstruction of complete time series fragmented into the reconstructions of series
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DrCuse
0.90
Rustimeo
SHD
080
635
639
logged Length
lagged Length
lageed Length
lagged Length
Figure A6: Results of DyCausal on datasets with maximum lag T 21 3. 5}.
W2
W2
09
0.
7.00
1794
0.-
04
Iagged Length
lagged Length
Ingged Length
lagged Length
Figure A7: Results of DyCausal on datasets with maximum lag
within sliding windows, which dilutes the negative effect of noise and improves the performance of DyCausal. In Figure[A8] we only show SHD less than 1000 and runtimes less than 2000; results beyond this range lose competitiveness. Since the runtimes of TECDI on graphs with 80 nodes are more than 12 hours, report the results with variance in Table A4 We can observe that DyCausal achieves the best and SHD in various models. In most settings, the TPR of DyCausal is also the highest. In addition, the result variance of DyCausal is very low, further demonstrating the robustness of DyCausal in various models.
only report the results of TECDI
graphs with 10, 20, and 40 nodes. We
C.10 ABLATION FOR THE ACYCLIC CONSTRAINT IN NONLINEAR SETTINGS
To demonstrate the necessity of improving the log-det acyclic constraint, we introduce ablation baseline DyCausal with hog. which uses the original log-det acyclic constraint without scaling the matrix through the matrix norm. We compare DyCausal and DyCausal with hiog on the causal graph with 10 nodes and nonlinear causality, where the maximum lag length = 2. Figure shows the ground-truth causal graph and the causal graphs estimated by DyCausa 1 and DyCausal with hiog. Figure A10 shows the trajectories of hnorm and fiog indexed by the number of iterations.
We can see that DyCausal not only perfectly identifies the causal graph, but also its acyclic con- straint value is effectively reduced as the iteration proceeds. In contrast, DyCausal with /o fails catastrophically. As expected, hog frequently exceeds the optimizable space during training, forcing the algorithm to roll back the model, reduce the learning rate, and retrain. As result, DyCausal with hiog runs over 60000 iterations, which is much more than the number of iterations DyCausal runs. In addition, due to the excessive reduction of the learning rate, the model cannot be effectively optimized identify causal structures. We also conduct experiments on the causal graph with 10 nodes and dynamic settings, where the maximum lag is 1. Figure All shows the causal graph estimated by DyCausal and DyCausal with doog when eoty runs successfully. Figure A12 shows the trajectories of hnrm and hing indexed by the number of iterations.
The results show that the use of hnorm can identify more accurate causal structures than that of hgog- Especially at both ends of the time series (W, and W DyCausal almost perfectly identifies the causal graphs, while DyCausal al with hoog struggles with redundant and missing edges. The trajectories of h norm and htgg indicate that hnorm decreases steadily early in training and
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14
1280
Runtlnnss
17%
SHID
38
Namber nodes
Number of nodes
Number of nodes
Number of nodes
(a) Results on linear structural equation models.
RyConad
NT5-0
4.4
Rusiimee
1200
TPP
SHTD
-50
42
Nomber nodes
Number of nodes
Nomber nodes
Nomber nodes
(b) Results on nonlinear structural equation models.
1753
774
SHD)
2
Number
Number of nodes
Nomber
Number uodes
(c) Results on Lorenz ordinary differential equation models.
Figure A8: Results on time series with static settings.
is closer to O at the end of training. The above results demonstrate that hnorm outperforms hogg whether in static or dynamic settings.
C.I1 SENSITIVITY ANALYSIS OF PENALTY COEFFICIENT
We conducted sensitivity analysis on the penalty coefficient 3 prove that our parameter setting is reasonable. We run DyCausal N 200 time series with length T 50 and number of variables (0.001, 0.005, 0.01, 0.05, 0.1).
50. Table A5 reports the performance DyCausal in 3
The results show that both too small and too large 8 can lead performance degradation, and DyCausal performs best when 8 0.05, which is consistent with parameter setting.
C.12 RESULTS ON NON-SMOOTH DYNAMIC CAUSALITY
In this paper, we assume that causal relationships change smoothly over time and use sliding Windows and coarse-grained approximations to learn dynamic causal relationships. However, we emphasize that DyCausal can adapt to non-smooth changing causal relationships. We discuss DyCausal's 's behavior in two cases. We set causal matrix as W = Woli.f]oost P regulates the frequency of weight changes, and E regulates the intensity of weight changes. In one case, we set 4, E = 2, window size K = and sliding step size 1. the other case, set P 1, F. 2, window size K and sliding step size S = 1. On N 200 time series
j.oos((ss ( H), where
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Table A4: Results on time series with static causality. The boldfaced and underlined results respec- tively highlight the methods that perform the first and second best among all compared methods. We omit the percentage signs % of TPR and F1 simplify the representation.
linear structural equation model
matrices
DyCause
DYNO
PCMCI
SVAM
TPR1
97.13:2.42
85.4616.95
95.59:1183
39.79-12..4
10 nodes
Fit
98.45+1.229
9..54++..4
58.87+-4.26
46.86+224.15
SHDL
1.80++1.54
9.30-+4.65
88.30=12.81
52.60=19.227
TPR1
97.11+2..43
93.38=3.79
96.26:11.67
65.35+6.84
20 nodes
F1t
98.52++2.43
96.54+2.06
49.37=t4.94
70.40=55.55
SHD
3.40-12.84
8.20-14.92
244.10:142.06
66.10:113.60
TPRI
96.87+1667
84.45=2.79
97.71
62.30+2.78
40 nodes
F11
08.40-88.77
87.25+2.85
44.35+43.69
70.57+
SHD4
4.906
36.50 6.45
599.10
125.00+66.99
TPRT
98.13+1.00
96.33+2.49
97.56++0.49
1.51+
80 nodes
F1f
99.06
98. 1.32
39.01
1.93+00.68
SHD4
9.00
17.60 12.02
1484.00 164.19
728. 428.50-46.06
nonlinear structural equation model
matrices
DyCausal
NTS-NO
PCMCI
SVAM
TPR1
92.35+3.12
93.33-55.49
78.05---4999
36.63+7..04
nodes
Fif
94.00-11.67 94.00=4.24
53.65:15.78
46.88
SHD
7.10:12.07
7.20-1422
82.10=17.166
49.20=19.91
TPR1
1..70
87.87
75.69-L4.08
38.5143.86
20 nodes
F1t
94.26:122.3
83.37-8.12
50.21114.29
51.27:14.7
SHD
63.80=16.35
45.90
183.70=222.29
90.10=10:44
TPR1
88.66
79.23 4.97
73.74
10.02+16.11
40 nodes
Fit
90.60-1..70
81.08
46,80113,29
13.33+21.05
SHD
12.30
90,20-124.83
413.40=49.07
300.50-171.72
TPR1
85.52.13.86
55.72+2.87
71.79:E1.29
1.13
80 nodes
FIf
90.09-12.35 70.75
41.40:E1.79
1.64
SHD4
90.40:21.72 219.67121.95
977.30
655.40
ordinary differential equation model
matrices
DyCausa
NGM
PCMCI
SVAM
TPR1
99.7510..75
91.75+2.51
83.7515..99
38.2543.72
10 nodes
FIt
09.55
91.90-2.116
61.2343.77
53.8743.84
SHD
0.20
9.30
49.701+5.60
31.90+3.44
TPR1
1..25.11.25
96.75+1.39
89.5013.633
41.6341268
20 nodes
F11
97.24196.09 96.0910.86
52.00:12.99
57.37+2.44
SHD
7.70
9.60=2.76
178.70=20.94
57.00
TPR1
94.30
99.38-0.88
88.44
42.56-2.559
40 nodes
F1t
94.30 1.67
95.34-1.29
32.1112.005
57.8442.38
SHD
32.20-19.22
10.30-8.20
823.10=14..11 14
117.60+8.79
TPR1
11.91
100.0010.00
85.00:12.74
43.31
80 nodes
F1t 94.13=11.16
11.4
19.56:+0.85
51.5442.25
SHD
63.50
114.00+21165
3148.30+94.59
359.70=31.79
with length 10 and number of variables 50, we validate the performance of DyCausal against non-smooth changes. The results are reported in Table A6
can be seen that in the first case, DyCausal accurately learned the causal graphs both ends and in the middle of the time series. However, we find that the causality within the sliding window does not change both ends and in the middle of the time series, which might be the reason why DyCausal accurately identified the causal graphs. Suppose that the causality changes from W to W' within the sliding window, where the sample size of the causality W is a and the sample size of the causality W is b. Observing the results, we find that the causality learned by DyCausal
within this window is wW
We define the window with changing causality transition win- 6+
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laga
lagı
instantaneous
Ground-truth
Estimated with Anorm
Estimated with hig
Figure A9: Results of DyCausal and DyCausal with htog 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 hiog-
hog
-3
log(8... log(h
-8-
-10
-6-
5000
10009
15000
20000
25000
31000
0
50000
10000
20000
30000
80000
60000
Number of itertations
Number of iterations
Figure A10: The trajectory of acyclic constraint values indexed by the number of iterations.
dow; otherwise, it is stationary window. DyCausal 1 can identify the changing trend of causality and accurately identify the causal graphs in the stationary window. In the second case, we set the window size to 1. Due to the small sample size within the window, DyCausal achieves a reduced performance, but still identified causal graphs with acceptable accuracy at each time step. Although the coarse- coarse-grained grained approximation no longer works when setting window size to 1, is not usual to change the causality window size K > as in the first case, identify the changing trend of the causality, and learn the accurate causal graphs in the stationary window.
frequently and strongly in this experiment. More often,
set the
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W;
0.5
1.5
2.0
W
Ground-truth
Estimated with frao
Estimated with Ave
Figure A11: Results of DyCausal I and DyCausal 1 with htog- The left is the ground-truth causal graphs, the middle the causal graphs estimated by DyCausal, and the right is the causal graphs estimated by DyCausal with hiog
2-
0-
log(hund
-6
lugthad
8
-00-
-4
12-
-14
5000
10000
15000
20000
5000
10000
15000
20000
Figure A12: The trajectory of acyclic constraint values indexed by the number of iterations.
C.13 RESULTS ON INSTANTANEOUS AND LAGGED CAUSALITY
To demonstrate DyCausal's 's ability to, respectively, identify instantaneous and lagged causal rela- tionships, we conducted experiments in dynamic settings. Specifically, we conduct DyCausal on N 200 time series and set the length of series and the number of variables (T 10, 20) and (T = 50, d = 50). The results are reported in Table A7
The results show that DyCausal accurately identifies the instantaneous and lagged causal relation- ships.
1. RESULTS ON REAL-WORLD DATASETS
We also run DyCausal on four additional real-world datasets, NetSim (Smith et al. 2011
DREAM-3 (Prill et al. 2010), Phoenix (Hossain et al. 2024) and CausalRivers Stein et al. 2025 to prove its generalization and adaptability to real-world issues. In particular, we chose CUTS+ and NGM as baselines for comparison. The former performs second only to DyCausal on the CausalTTmee dataset, and the latter models ordinary differential equations to identify causality, which are often used time series in the biological field. We record AUROC in Table A8 to estimate their performance.
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Table A5: Results of sensitivity analysis on B.
0.001
3
0.005
/
01.01
/
0.05
0.1
TPR
86.07
88.91
77.09
93.65
55,95
W
F1
66.46
77.36
70.02
95.49
63.68
SHD
99.0
51.5
70.0
9.0
55.0
TPR
92.87
95.78
98.95
98.93
79.06
W 25
F1
88.37
94.77
98.43
99.46
86.83
SHD
24.0
10.0
3.0
1.0
63.0
TPR
81.48
87.91
79.65
95.09
48.66
W 50
F1
76.20
74.52
77.36
96.06
59.08
SHD
54.0
59.5
59.5
8.0
63.0
Table A6: Results on non-smooth dynamic causality.
P =
4, E =2
P= 1,
E
=
W1
W4
Ws
W1
W2
W3
W4
W:
W8
Wr
Wa
W9
W 10
TPR F1
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
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
We observe that CUTS+ achieves the lowest AUROC on NetSim. This may be because the data generation process of NetSim is more consistent with the ODE model. Therefore, NGM achieves better performance than CUTS+ on NetSim. Both DREAM and Phoenix are datasets about gene expression levels, where DREAM contains 100 genes and Phoenix contains 11165 genes. Due to the large scale and the lack of a verifiable causal network, we screen 34 transcription factors based on the gene regulatory network provided by Phoenix and take the regulatory network of these transcription factors as the target of causal discovery. For CausalRivers, we use the default sub- dataset Random 5. This sub-dataset contains randomly selected river nodes and provides the flow time series and causal graph of these nodes. The results show that DyCausal achieves significantly higher AUROC than the baselines on all four datasets. This is attributed to flexible adjustment of the reconstruction model to adapt to different datasets.
D IDENTIFIABILITY OF DyCausal
Identifiability is the most fundamental issue in causal discovery, and in this section, we discuss identifiability DyCausal. Looking back at the process learning causal structures, we reconstruct the data in the window [t, + K] optimize the causal structures at the time step t. Although this process slightly violates the dynamic causality (causality at time f is only responsible for generating observations at time t). assume that K is small enough and that causality is approximately invariant within the interval [t. + K). Therefore, identifying dynamic causality across the entire time series is broken down into identifying approximately static causality within multiple sufficiently small intervals. According to the identifiability results in (Shimizu et al.) 2006 Peters Bühlmann, 2014 Hoyer al. 2008), when the sample size tends infinity, no other DAG can generate the same distribution. Therefore, when observed data within the interval [t, + K] are generated by the structural equation model defined in Eq. (I) and the observed sample size of the time series is sufficient, we can identify the Markov equivalence class of the causal graph and restore the coefficients of the causal matrix W, time t. Furthermore, based on the identified coarse-graines grreeegg causal matrices, restore fine-grained causal matrices through linear interpolation. According to Eq. (5). the dynamic causal matrix a continuous function of the time index, we can approximate the causal matrix at each time step by linear interpolation within sufficiently small interval [t, + S]. In summary, dynamic causality identifiable for DyCausal.
E THE USE OF LARGE LANGUAGE MODELS
We promise that we did not use large language models (LLMs) in this work.
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Table A7: Results of instantancous and lagged causality.
results of instantancous causal graphs
T'
50, d
50
T
10,
20
W
25
W00
W1
2
W3
W4
Ws
We
W7
W
W9
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 lagged causal graphs
T' =
50, d = 50
T
10, d
20
W
W 25
W50
W1
W2
W
W
W
Wg
W7
Wa
8
Wg 9
W 10
TPR
94.38
96.97
92.92
90.81
86.
90.85
84.25
93.16
93.13
84.02
87.56
86.67
90.28
F1
96.67
98.30
95.50
93.50
91.57
94.79
91.12
95.75
96.32
90.50
92.29
91.84
92.43
SHD
6.0
3.2
7.6
4.8
8.4
5.9
7.5
2.5
2.1
8.3
8.7
8.3
5.7
Table A8: AUROC on NetSim, DREAM-3 and Phoenix.
NetSim (d 15)
DREAM (d 100)
Phoenix (d 34)
CausalRivers (d 5)
DyCausal CUTS+
0.9391=0.000
0.0196
0.5472-0.0185
4.7214410.0400
0.7058=000071
0.6229-:0.0138
0.5104-0.0151
0.6072++00027
NGM
0.7090-0000000
0.5629-10.0000
0.4809-0.0083
0.4905-00.052
27


Reviews:
[{"review_id": "GOBPCZATwX", "reviewer": "Reviewer_wtX2", "summary": "This paper proposes structure learning for dynamic (i.e., time-varying) causal graphs in time series. This is proposed via a \"coarse-to-fine\" strategy; this proceeds by roughly identifying causal structures over some large sliding window, then linearly interpolating. An additional modification to the acyclicity penalty of DAGMA is introduced. Extensive experiments are undertaken, which show superior performance of the proposed method in real and synthetic datasets with time-varying causal mechanisms.", "questions": "1. How does h_log compare with more extensive hyperparameter tuning? Do the results of C.10 change? Furthermore, does the proposed version h_norm significantly change results in the general (static) structure learning setting?\n\n2. How is initialization performed? Dagma initializes W to be the zero matrix, but this obviously results in an infinite denominator.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper is well-written and easy to follow.\n- The paper motivates and undertakes an important and under-discussed problem of time-varying causal structure learning.\n- Relative to some prior work (e.g., DyCast), which only learns time-varying instantaneous graphs, DyCausal simultaneously learns time-varying lagged relationships. This seems to significantly improve performance. \n- The experiments clearly demonstrate strong performance relative to relatively recent and strong baselines.", "weaknesses": "I'm not totally convinced by the analysis of the revised log-determinant penalty. One reason why is that the benefits of DAGMA lie partially in nice optimization properties in a barrier method approach -- it's a priori possible but not obvious to me that the proposed version would perform better in the general structure learning setting. The ablation in C.10 is nice in theory, but the graph in Figure A9 seems to moreso suggest some catastrophic failure than poor optimization.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "SWyYcAyLbY", "reviewer": "Reviewer_DRWo", "summary": "The paper proposes DyCausal, a framework for learning dynamic causal structures in time series that exhibit complex and time-varying dependencies commonly observed in real-world systems. The method addresses fully dynamic causality, where both instantaneous and lagged dependencies evolve over time. To tackle this challenge, DyCausal leverages convolutional networks to capture causal patterns within coarse-grained time windows, and applies linear interpolation to refine the causal structures at each time step, thereby recovering fine-grained and continuously evolving causal graphs. In addition, an acyclicity constraint based on matrix norm scaling is introduced to enhance computational efficiency while effectively preventing loops in the evolving causal structures. Experimental results demonstrate the effectiveness of the proposed approach.", "questions": "1. What are the limitations introduced by the coarse-grained approximation? \n\n2. Can the authors establish theoretical guarantees for convergence of proposed framework, similar to the Lemma 6 in [1]?", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 1, "strengths": "1. The paper extends causal structure learning to a general setting of fully dynamic causality, where both instantaneous and lagged dependencies evolve over time. This broadens the applicability of causal discovery methods to more realistic time-varying systems.\n\n2. The proposed approach appears technically sound.", "weaknesses": "1. The paper lacks significant technical innovation. Most components of the proposed framework appear to be incremental adaptations of existing techniques rather than conceptually new contributions. The use of convolutional networks for coarse-grained temporal modeling and the acyclicity constraint are engineering extensions that do not substantially advance the methodological frontier. Moreover, the theoretical results are mostly straightforward derivations of [1], without introducing new analytical insights.\n\n2. The presentation and notation require improvement. For example, in Theorem 2 the symbol $=\\in$ is incorrect; in Algorithm 1, the L₁-norm notation is inconsistent; and the usage of boldface for matrices and vectors is not uniform (e.g., the identity matrix I on line 214). In addition, the definition of Δ in Theorem 1 (line 239) is unclear, and line 7 of Algorithm 1 is difficult to interpret. These issues hinder clarity.\n\n3. The experimental section contains several marking or annotation errors. For instance, on line 1098, the reported TPR: NTS-NO better than DyCausal. Similarly, see line 832.\n\n[1] Bello K, Aragam B, Ravikumar P. Dagma: Learning dags via m-matrices and a log-determinant acyclicity characterization[J]. Advances in Neural Information Processing Systems, 2022, 35: 8226-8239.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "vpHSybTdQv", "reviewer": "Reviewer_QNBN", "summary": "This paper proposes a coarse-to-fine causal discovery method to address fully dynamic causality. They leverage CNN to capture patterns within coarse-grained time windows, and apply linear interpolation to refine causal structures. The authors tested DyCausal's performance and compared it with previous methods on both synthetic and real-world datasets.", "questions": "1. There seems to be a lack of sensitivity analysis of penalty coefficient $\\beta$, decay coefficient $\\mu$, and decay rate $\\gamma$.\n\n2. The model uses a sliding window approach for causal discovery. The question is whether the causal graphs obtained at the intersection of two windows are consistent. For example, is there a large difference in the changes between $W_{t+S}$ and $W_{t+S+1}$? Are they smooth?\n\n3. The experimental results on CausalTime do not appear to include some of the latest state-of-the-art methods, such as JRNGC.\n\n4. Are there any experiments on a real-world dataset with large-scale nodes?\n\n5. As far as I know, DYNOTEARS and NTS-NOTEARS are both static causal discovery methods. How do they obtain the results on the dynamic dataset?\n\n[1] Zhou, W., Bai, S., Yu, S., Zhao, Q., & Chen, B. (2024). Jacobian regularizer-based neural granger causality. arXiv preprint arXiv:2405.08779.", "limitations": "", "rating": 4, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The authors addressed the limitations of previous methods by considering a fully dynamic causality.\n\n2. They offered clear motivation for their DyCausal algorithm and provided a new form of acyclic constraint.\n\n3. The authors tested DyCausal's performance and compared it with previous methods.", "weaknesses": "1. The proposed refined dynamic causal graph interpolation is based on the assumption that the causal relationship changes smoothly and follows a linear law, which is a very strong prior assumption.\n\n2. The acyclicity constraint in the question appears to be a variant of the DAGMA constraint, and I am skeptical as to whether it is sufficient to be considered an innovative point.\n\n3. Learning causal relationships requires theoretical guarantees of identifiability. The proposed CNN method employs an encoder and decoder to infer causal structures, treating it as an unsupervised sequence reconstruction task, which appears to lack theoretical support.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "j5tySJIKlR", "reviewer": "Reviewer_YHJh", "summary": "This paper introduces a novel method for time-series causal discovery, specifically designed to handle dynamically changing causal relationships. This is a timely and important problem that has received relatively little attention in the literature. The proposed approach is well-motivated, and the paper is clearly written and supported by extensive empirical evaluations.\n\nWhile the work is promising, several points regarding the experimental validation, comparison to related work, and clarity of the evaluation protocol require attention. Addressing these points would strengthen the paper and allow me to raise my rating further.", "questions": "Answering these questions in the paper or rebuttal would greatly improve clarity:\n\n1. **Robustness to Non-Cyclic Dynamics:** What is the expected behavior of the model if the underlying causal dynamics are not smooth or cyclic, but instead feature abrupt changes or non-periodic patterns?\n2. **Acyclicity Constraint:** Could the acyclicity constraint be relaxed or removed? If so, what would be the expected impact on performance?\n3. **Performance on Static Graphs:**  Can you give a rational why your method is typically even better than other methods in the static case? (A8). This suggests some Hyperparameter problems.\n4. **Interpolation vs. Per-Step Estimation:** Do you have to perform linear interpolation, or is this something you simply do for computational efficiency? If this is the case, how do results change if the Causal Graph is estimated at every time step?\n5. **Dynamics of Instantaneous Links:** There appears to be a contradiction between Eq. 2 (which suggests instantaneous links are static) and Section 5.1 (where they seem to be varied in experiments). Am I misunderstanding something?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- **Clarity and Presentation:** The paper is well-written and mostly easy to follow. \n- **Novelty and Timeliness:** The paper addresses an important and under-explored problem in causal discovery: learning dynamic causal graphs from time-series data. This represents a valuable and timely contribution.\n- **Empirical Evaluation:** The authors provide an extensive set of experiments on both synthetic and real-world datasets, which helps to demonstrate the method's effectiveness across various settings.", "weaknesses": "### **Major Weaknesses**\n\n1. **Clarity on Real-World Dataset Dynamics and Evaluation:** The justification for using certain real-world datasets (e.g., `NetSim` or `CausalTime`) to evaluate a *dynamic* causal discovery method is unclear. To the best of my understanding, these datasets have a static ground truth graph (as they rely on some synthetic components).\n    - The authors should clarify why a dynamically changing graph is expected or found in these cases. If there is no prior information on dynamics, this should be explicitly stated, as it impacts the interpretation of the results.\n    - To further validate the method's ability to handle known dynamics, the authors are strongly encouraged to evaluate on benchmarks like **CausalRivers** [6]. This dataset's assumption of a fixed graph structure with dynamically changing edge weights perfectly matches the problem setting of this paper and would provide more convincing real-world evidence.\n\n2. **Usage of $h_{norm}$:** While Eq. 7  suggests that the norm is used as a constraint, Algorithm 1 shows that it is simply added as a loss term. This suggests that it is not enforced that $h_{norm}$ is exactly 0, but rather it is only pushed close to 0 to minimize the loss. As the DAG structure is only enforced at $h_{norm} = 0$, how is it guaranteed that the solution is really a DAG? Please clarify this point.\n\n3. **Ambiguity in Evaluation Protocol:** How are the performance metrics in Table 1 calculated for dynamic graphs? For instance, are the predicted graphs over time collapsed into a single summary graph for comparison with a static ground truth? How are baseline methods that produce only a single, static graph treated in this evaluation? This information is essential for interpreting the results correctly.\n\n4. **Lack of Architectural Comparison and Positioning:** The paper would be strengthened by a more detailed comparison with related convolutional approaches such as NTS-NOTEARS, TCDF, and other recent works (e.g., [4, 5]). A discussion of the architectural differences and how they relate to the modeling of temporal dependencies would help readers better understand the unique contributions and advantages of the proposed method.\n\n5. **Need for Granular Performance Analysis:** The paper would benefit from a more fine-grained analysis of the model's performance. Specifically, providing separate recovery scores for **instantaneous** and **lagged** causal links would offer deeper insights into the method's strengths and weaknesses.\n7. **Unexplained Superiority on Static Graphs (Appendix A8):** A key result that requires further explanation is the proposed method's superior performance even on static graphs. This is counterintuitive and could suggest suboptimal hyperparameter tuning for the baseline methods. A more thorough investigation or discussion is needed to justify this finding and rule out experimental artifacts.\n\n\n\n### **Minor Weaknesses**\n\n- **Assumption on Graph Structure:** The method appears to assume a fixed graph structure with dynamically changing edge weights. While not a significant weakness, this should be made clear earlier in the paper, as the method does not address the problem of a changing graph *structure* (i.e., addition/removal of edges).\n- **Smoothness Assumption:** The paper could benefit from a brief discussion on the limitations of the smoothness assumption for the function `F`. What happens in natural systems with phase transitions or abrupt changes (e.g., freezing at 0°C), where this assumption is violated? A small synthetic experiment on this (e.g., simply drawing coefficients from a uniform distribution) would be very interesting.\n- **Literature Review:** The background section mentions two groups of CD methods, which may be an oversimplification. The review could be extended to better position the work by including other major families of methods, such as Granger Causality and constraint-based/noise-based approaches [1].\n- **Related Work:** A recent paper at ICLR, on Meta Causal Graphs [2], appears closely related and should be discussed.\n- **Terminology:**\n    - **L49 \"Faithful\":** The use of the word \"faithful\" could be confused with the Causal Faithfulness assumption, a standard term in the causal inference literature. Rephrasing is recommended.\n    - **\"Significant\":** The word \"significant\" is often used without statistical validation. Please ensure its use is appropriate or rephrase to be more descriptive (e.g., \"substantial,\" \"large\").\n    - **SEM vs. SCM:** The paper uses the term Structural Equation Model (SEM). To avoid confusion with the large body of work in social sciences also using this term, consider using Structural Causal Model (SCM) instead, as suggested by [3].\n- **Clarity and Typos:**\n    - **Table 1:** Typo `CUT+` should be `Cuts+`.\n    - **Figures:** Many figures use labels like `W1`, `W2`, `W3` without a clear definition in the caption or in the legend. Adding a brief explanation in the figure captions would improve comprehension.\n    - **Precision Metric:** The precision values in the tables appear to be between 0 and 100 but are reported without a decimal point (e.g., \"95\" instead of \"0.95\"). This should be clarified in the table captions.\n\n\n\n¹  https://www.jair.org/index.php/jair/article/view/13428/26917\n\n² https://openreview.net/forum?id=J9VogDTa1W\n\n³ https://library.oapen.org/bitstream/id/056a11be-ce3a-44b9-8987-a6c68fce8d9b/11283.pdf\n\n⁴ https://arxiv.org/abs/2408.08023\n\n⁵ https://arxiv.org/html/2404.01466v1\n\n⁶ https://openreview.net/forum?id=wmV4cIbgl6", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)