Physics-Aware Spatiotemporal Causal Graph Network for Forecasting with Limited Data
Authors: Zijun Cui
Sam Griesemer
Sungyong Seo
Joshua Hikida
Yan Liu
Venue: Accepted by TMLR ()

Abstract: Spatiotemporal models have drawn significant interest recently due to their widespread applicability across many domains. These models are often made more practically useful by incorporating beneficial inductive biases, such as laws or symmetries from domain-relevant physics equations. This "physics-awareness" provides an interpretable means of grounding otherwise purely data-driven models, improving robustness and boosting performance in settings with limited data. In this work, we view physical dynamics as domain knowledge that captures fundamental causal relationships across space and time, and can be effectively leveraged by our proposed physics-aware spatiotemporal causal graph network (P-STCGN). We firstly describe a means of deriving causal relationships from spatiotemporal data, serving as physics-aware labels to learn a causal structure via a dedicated neural module. We then formulate a forecasting module that can operate under this causal structure, producing predictions that are guided by physics-aware cause-effect relationships among modeled variables. Extensive experimentation demonstrates that our method is robust to noisy and limited data, outperforming existing models across a variety of challenging synthetic tasks and benchmark datasets. We further evaluate our method on real-world graph signals and observe superior forecasting performance, achieved by effectively utilizing causal signals from prior physics knowledge.

Published in Transactions on Machine Learning Research (07/2025)
Physics-Aware Spatiotemporal Causal Graph Network for Forecasting with Limited Data
Zijun Cui
cuszijun@msu.edu
Michigon State University
Sam Gricsemer" University of Southern California
semgriesemmer&wusceede
Sungyong Seo*
nungyongs@uscc.el
University of Southern California
Joshua Hikida
University of Southern California
Yan Liu
yonliu. es www....cc...e edu
University of Southern California
Reviewed on OpenReview: https: //
// openreview. net/ forum? id=n3yrVaPclla
Abstract
Spatiotemporal models have drawn significant interest recently due to their widespread applicability across many domains. These models are often made more practically useful by incorporating beneficial inductive biases, such as laws or symmetries from domain-relevaan physics equations. This "plysics-awareness" provides an interpretable means of grounding otherwise purely data-driven models, improving robustness and boosting performance in scttings with limited data. In this work, we view physical dynamics as domain knowledge that captures fundamental causal relationships across space and time, and can be effectively leveraged by our proposed physics-aware spatiotemporal causal graph network (P-STGNN. We firstly describe means of deriving causal relationships from spatiotemporal data, serving as physics-aware labels to learn causal structure via a dedicated neural modulc. We then formulate a Torecasting module that can operate under this causal structure, producing predictions that are guided by physics-aware cause-effect relationships among modeled variables. Extensive experimentation demonstrates that our method is robust noisy and limited data, outperforming existing models across a variety of challenging synthetic tasks and benchmark datasets. We further evaluate method real-world graph signals and observe superior forecasting performance, achieved by effectively utilizing causal signals from prior physics knowledge.
1 Introduction
Spatiotemporal modeling has drawn significant interest recently due its wide application in climate, traffic systems, electricity networks, and many other fields. Complex machine learning models (e.g., deep neural networks) achieve superior performance in data-rich settings, such as computer vision, natural language processing, etc. Employing these models in spatiotemporal settings, however, poses set of new challenges. The training data are often reooutios,,,,,nn, of widely varying quality, and heavily constrained by physical principles. Further, real-world applications are typically characterized by a limited amount of training data. To address these challenges, integrating domain knowledge with data-driven models has emerged as one of the most promising directions forward, clearing path for the construction of more robust and interpretable

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Published in Transactions on Machine Learning Research (07/2025)
pipelines. Specifically, physics-informed or physics-aware machine learning is of significant importance for spatiotemporal modeling as observations in the physical world (e.g,, meteorological measurements and traffic flow) are naturally governed by physical principles.
Meanwhile, an emerging topic in deep learning community is causal inference and analysis in time series (Runge 2018 Runge al., 2019aa Nauta et al.] 2019; Runge et al.) 2019b). For example, [Pamfil al. (2020) introduce smooth acyclicity constraint to multivariate time series inspired by (Zheng et al. 2018]. Amortized causal discovery (ACD) considers an explicit separation between causal structure learning and the
downstream dynamic prediction task (Löwe et al. 2022). While many works discover causal structures directly from observational data, they often assume data sufficiency, requiring numerous samples for accurate causal discovery and being sensitive to data noise. Physics equations naturally connects to causality, rooted in the fundamental principle that an effect can not occur before its cause. Observations from the same dynamic system, even when driven by different underlying physics equations, share common causal relations across space and time that are intrinsically present the dynamics.
For example, when the heat equation ( (hu DÀu where D is diffusivity constant) is considered, we know 35
that temporally first order and spatially second order derivatives are involved. We then specify causes and effects on a discrete domain (time interval At) as:
u,(1 + = (1) + AL. DAu
= (t) + At- D
Σ (u,(t) - (t)), JEN.
(1)
where A is the Laplace operator and M2 is a set of adjacent nodes of i-th node. Eq. [ shows the discrete Laplace operator. For the target value (t 1), the variables in the right-hand side are regarded known causes from the heat equation. In the physical realm, the spatiotemporal observations, such the climate and weather measurements (Kashinath et al. 2021). inherently adhere to physical principles. Physics equations are thus typically acknowledged as valuable information for robust spatiotemporal modeling. Nonetheless, the exploration of physics-aware causality remains scarce.
In this work, we introduce a novel physics-aware causal graph network (P-STCGN), unveiling physics-aware causality in dynamic systems. In our modeling process, we decouple causal structure learning from dynamic forecasting. Physics-aware causality is derived from prior physics knowledge. A causal module is introduced to learn causal relations from analytically derived physics-aware labels, such that it capture the causal structures aligned with the physics laws. With the learned causal structure, forecasting module learns hidden representations with corresponding causes to predict effects. Our main contributions are:
We propose an innovative physics-aware causal structure learning approach. A causal module learns relationships given additional explicit labels extracted from physics knowledge.
We present a novel physics-aware causal graph network that harnesses the insights derived from prior physics knowledge within a causal framework.
We demonstrate the robustness of our physics-aware causal learning approach in handling noisy and limited data. It can be further extended to handle hidden confounders. Moreover, P-STCGN enhances forecasting performance over real-word graph signals and excels in terms of data efficiency and generalization.
2
Related work
Physics-aware learning Physics-informed learning is an emerging research direction where physics knowledge as strong inductive biases is utilized in the construction of interpretable and robust deep models (Meng
ct al. 2025). Different approaches have been proposed to incorporate physics knowledge into deep models, such model architecture design. de Bezenac et al. (2018) showed how the design of data-driven model be motivated by the advection-liffusion equation to predict sea surface temperature. Seo et al. 2019
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Time
Time H+
Figure 1: Heat dissipation over 2D space and time. Nodes in the graph structure correspond to sensors and the observations at each sensor are time varying. Given the heat equation (ú = Dåu), we can provide spatial (blue) and temporal (green) causal relations from previous nodes to a current target node (white).
constructed the proposed architecture inspired by general form of PDEs physical systems. Employing physics principles as training loss is another widely adopted approach, such as the Hamiltonian neural network (Greydamus et al. 2019). Physics knowledge has been used for spatiotemporal modeling since usually governed by underlying dynamic mechanisms. Kaltenbach & Koutsourelakis (2021) propose a novel physics-aware generative state-space model for long-tern term predictions, and Wang et al. (2020) combine two turbulent flow simulation techniques with deep neural networks to predict physical fields.
dynamic observations
Despite these achievements, physics-aware causality is not very well studied in this context.
Causal discovery in time scries Discovering underlying causal structure in time series data a fundamental
problem that remains actively studied loday (Runge, 2018 Runge al. 2019a Nauta al., 2019 Runge
et al. 2019b Rubin [1974]; Pearl (2099) Imbens & Rubin (2015) introduce the problem and provide
mathematical framework for causal reasoning and inference under causal graphical models (also known
Bayesian networks (BN)) (Koller & Friedman 2009). Granger (969) formalizes concept of quantifiable
causality in time series, called Granger causality. Learning causal associations from time series is also an emerging topic in the deep learning community. Runge (2018) proposed a method to distinguish direct from
indirect dependencies and common drivers among multiple time series reconstruct a causal network. Rungee et al. 2019b) quantify causal associations in nonlinear time series and Runge et al. (2019a): Nauta et al. 20019
provide promising applications of causal discovery in time series. Pamfil et al. (2020) introduce smooth acyclicity constraint to multivariate time series inspired by (Zheng et al. 2018) who consider causal discovery purely continuous optimization problem. Amortized causal discovery (ACD) is considered an explicit seperation between causal structure learning and the downstream dynamic prediction task (Löwe et al.) 2022). Although many works are capable of discovering unknown causal structures from obscrvational data directly, they usually assume data sufficiency, i.e., sufficient samples are available for accurale causal discovery. Besides, the performance can be sensitive data noise. In contrast, we consider physics-avee awarre causal discovery approach. We leverage explicit causal relations inferred from domain knowledge physics-aware causality for robust classification and retrieval under limited and noisy data.
GNN-based spatiotemporal modeling. While many graph- architectures have been proposed to handle spatiotemporal observations on irregular domains (Li et al.) 2018 Wu et al. 2020 one of the core
issue behind graph- graph-based neural networks (GNNs) (Hamilton et al.
2017
Kipf Welling 2017 Battaglia
ct al. 2018 Zhu al.) 2020) is that they cannot distinguish node-wise differences. To address this limitation,
attention mechanisms have been extended to GNNs (Veličković al. 2018 Zhang et al.) 2018 Zheng et al., 2020). (Serrano & Smith 2019), however, shows that attention weights do not necessarily correspond to importance for an output, but are instead more likely to be noisy predictors. Hence, there is need for incorporation of prior knowledge into the unspecified representations between two nodes for reliable learning. Many graph-oriented methods leverage lask-specific formulations (e.g., [Zheng & Zhang (2023) for traffic, Zhang et al. (2020) for gesture recoguition, etc) which can be effective, but sacrifice generality for well-tuned inductive biases. Physics-aware causality can be a direct solution to these challenges, being an effective way to dynamically incorporate problen-specific knowlodge (c.g., spatiotemporal relations corresponding arbitrary physics equations) into general graph neural network architectural base.
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STGC
Prior PDEs
(egg, Navier-Stokes equation, etc.
1
Module
Causal
W
W
F(ú, Qu Vu, ".
t-K+2
Physiswaaare Causalty
structure (DAGS)
STG
STGF
encoder
Spatial
Temporal encoder (FC)
Forecasting Module
Prediction black
-
1)
(GConv)
(u,(t) NV
Figure 2: An overview of the proposed physics-aware spatiotemporal causal graph network (P-STCGN). A sequence of graph signals firstly fed into a spatiotemporal graph network (STG), composed of a graph convolutional (GConv) spatial encoder and fully-connected (FC) temporal encoder. This is followed by two subsequent modules: (1) physies-aware causal module (STGC), and (2) spatiotemporal forecasting module (STGF). Prior PDEs from physical principles provide physics-aware causality. The red arrows denote how the supervised objectives are defined.
3 Problem formulation
Given observational data XI,... XT, where X; R.ND., assunne that static graph G. (V., E.) is given (or can be constructed by features of each variable). The nodes V. correspond to X, E RxD,, (N nodes each associated with a Dy-dimensional observation vector) and the structure &, is shared across different timestamps. With different nodes, the observations X E R TxNx can be regarded as a multivariate time series.
Additionally, we assuine the existence of prior knowledge that could be beneficial for modeling the observations. This is mild assumption since real-world observations are usually governed by physical principles, such meteorological measurements obtained from sensors in an automatic weather station (AWS). For example, suppose set of observations originate from weather sensors like those in an AWS: we expect domain-specific knowledge or equations related to weather phenomena to be beneficial for understanding the dynamics of the underlying weather system, and ultimately guide a learning method toward improved forecasting performance and generalizability. One particularly important prior equation for turbulent dynamics is the Navier-Stokes equation (Wang et al. ). These equations can be commonly represented as a function of spatial and time derivatives F(ù, ... Vu, v²u,...) O, where á and i denote the first and second-order time derivatives, respectively, and V represents the operator for the spatial derivative. As the continuous operators can be numerically decomposed in a discrete domain (c.g., onto graph structures), we can explicitly define causes for a larget observation at time / and extract causal relations accordingly. Note that causal relations derived from a particular equation are only partially complete due to the uncertainty surrounding the true governing oquation. The available prior knowledge need only be partially relevant to the underlying dynamics in order be beneficial.
Given the physics-awand aware causal relations, maximum time lag for causality. In the length K observations X,-K-I, X,, there are NK total mutually correlated observations, and we define N. causal relations among the NK x NK possible relations. In Fig. D we have N nodes in Gs, and the total number of variables in the length K sequence is 10. Thus, there are 100 possible relations between the 10 variables, and the Heat equation (Eq, D elucidates N. = 13 (5 temporal and 8 spatial) causal relations. We denote the causal graph 9. = (Vex & where [Vl NK and [.] Ne. Given the physics-awes causal relations (G.) derived from prior physics knowledge, our task is find model:
assign explicit labels between NK variables, where K is
X F(X.KKI,," X.; G.;9,, O).
(2)
where θ is set of learnable parameters in model F(.)-
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4 Proposed model
We describe the details of our proposed model, namely Physics-aware Spatiotemporal Causal Graph Networks (P-STCGN). The P-STCGN employs a two-stage learning approach to explicitly decouple causal structure learning from dynamic forecasting. Fig. _ shows a high-level view of P-STCGN consisting of two key modules: (1) physics-aware causal module (STGC), and (2) spatiotemporal forecasting module (STGF). STGC learns causal structure with causal labels derived from physics equations. Through STGC, we integrate the inductive bias into model through semi-supervised causal structure learning. STGF then performs forecasting tasks using the learned causal structures. Both networks are designed to learn node reprosentations from spatially and temporally correlated obscrvations.
4.1 Physics-awand aware causality
Given equation that is moderately beneficial for understanding the target dynamics, we can always define causal graph given a physics equation, by decomposing the equation into causes and effects analytically. One particularly important prior equation for turbulent dynamics is the Navier-Stokes equation (Wang
2020). Consider a fluid dynamics system governed by the Navicr-Stokos equation. The causal graph Go V, is defined as follows:
ctt al.
Nodes (V): Each node (i,j) represents spatial location in the 2D fluid domain. The variables each node include the velocity vector (t) and pressure (t. at time t.
The edges in the graph capture the causal relationships between nodes. Specifically, for each node (i,j), the Navier-Stokes equation implies the following relationship for the velocity vector:
Edges (E):
At -' u u (t+(+1)) (.
(t) At u...yuu
P
Discretizing the spatial derivatives using central differences on two-dimensional grid (x, v), we get the following equation:
V (l, Ardy (1+1,(()++ + u=-1,(()+ (::
+ (1)+((() + ui.ji(() (1)
(4)
where Ar, Ay the grid spacing.
This causal graph captures the dynamic interactions between different spatial locations in the fluid system according to the Navier-Stokes equation. We can also define the causal graph for the heat equation ăr DAu. These equations can be commonly represented as a function of spatial and time derivatives
8u
F(ú, ü. ... Vu, V -.=...)))
(5)
where ú and ü denote the first and second-order time derivatives, respectively, and V represents the operator for the spatial derivative. Discretization of continuous operators (e.g., over graph structures) enables to explicitly define causes for a target observation at time and extract causal relations accordingly. We further present. a theorem to ensure the acyclicity of the defined causal graph given physics equations below. Its proof is included in Appx.
Consider a physics system described by set of equations E that capture dynamics. Let G. E) be the causal graph derived from these equations, where nodes V represent variables, and directed edges represent. causal relationships. If the physics equations in E are characterized by the following properties:
(a) Linearity The equations are linear, and no variable appears with nonlinear dependencies, expressed as
du, = Earu, + be,
Vi
dt
where u, is the i-th variable, az are coefficients, and b, represents external influences that are independent of
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(b) Causality preservation Each variable's rate of change is solely determined by the values of variables in its causal neighborhood, including all immediate causes that affect the rate of change Mi, expressed as
du; fi(ueausal) + gi((),
dt
where Weansal represents the endogenous variables influencing ui, and gi(1) represents explicitly modeled exogenous influences.
(c) No instantaneous feedback There are no instantaneous feedback loops where variable directly influences itself in the same time step, expressed as
dug di + hy (ui,t))
Vi
Then, the causal graph G. is acyclic.
These three properties (linearity, causality preservation, and the absence instantaneous feedback) are commonly satisfied by physics equations. In particular, causality is inherently preserved in classical physics well in special and general theories of relativity (Riek & Chatterjee 2021 The causal relations exist in the defined causal graph serve physics-aware causal "labels" for causal structure learning, as introduced in the following.
4.2 Architecture
We first learn node-wise latent representations by two modules: a spatial encoder (SE) and a temporal encoder (TE). Spatial encoders are designed to learn spatial dependencies at each timestamp via the static graph structure G.- The spatial encoder generates K different snapshots which are grouped and fed into the temporal encoder as follows:
(S. = SF( 9.) 1'==t -+,, ,0)
(6)
Ze = TE((S--P. 1 ft K ,t)),
(7)
where Ze E NxD. is a set of node representations (dimension Dz) at time 1'. P is an aggregation order and TE merges the current embedding Sy and past P embeddings Sy- for spatiotemporal node embeddings f. This temporal encoder does not consider the graph structure.
Physics-aware causal module (STGC) Once node embeddings obtained, two D. dimensional vectors are fed into causal module (CM), which computes a probability of association between the two corresponding nodes:
2/ BCM(M, (イ Zt..):
(8)
where CM is fully-connected network and Ze, is the j-th node's representation at time t,. Since there are N different nodes at each time (with a total of K different timestamps), there are 22 different settings for p. If observations are stationary and the causal relations are independent on the absolute timestamps (ty, 4)), but dependent on the relative time interval r = f, t,, Eq. can be reduced pJ. = CM(Z Ze..).
It is worth noting that the causal module does not explicitly aim to recover some verifiably correct causal structure per se. Instead, it incorporates useful signal present in the graph Ge, which itself represents causal links across time and space as determined by available governing physics equations. We therefore use the resulting "probability of association" " D)." between two nodes as a preay for a causal link, and references to outputs of the causal module hereafter should be taken as implicitly incorporating this fact (i.e., that the CM embodics a useful representation of causal prior). This facilitates more flexible means of integrating domain knowledge (compared to strictly enforcing some Ge, say), as practice, the equations from which the causal graph is defined may be incomplete or outright incorrect.

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Spatiotemporal forecasting module (STGF) This module is used to learn node representations from spatiotemporal observations. It takes the learned causal structure from STGC and is used for the prediction of future signals. We introduce the forecasting module (FM) to transform the spatiotemporal representations Z to task-spocific representations. As CM learns causality- learn predietion-associaleed representations.
representations, FM is adapted to
H (M) I =t K K ,t),
where He E RNXD,, Since the causal relations from the NK past variables to N variables are inferred from "
STGC, the causal probabilities 2 (Eq. 8) are combined with H to predict next variables. Specifically, the
output H from FM in STGF and ş from CM in STGC are used to predict the next value at a node i and time ++1. 1:
X Σ
E
D Pjs H
(10)
K=t-K+IjEN
It's worth noting that we use causal probabilities between f' =tt-K+1,t-1 and instead of E [t- K + 1, and t ++1 1. There are two reasons for this: (1) since X is not available, it is impossible to compute 8 s+1 (a function of X(+1) in advance, and (2) we assume that the causality is stationary and thus from I' and is invariant if r = t- t' is unchanged. The second assumption is particularly valid for spatiotemporal observations in physical systems as most of physics-based phenomena are not dependent on the absolute time but relative time intervals.
Total Objective Function The total loss function L consists of the causal structure loss L. and the forecasting loss L,, as shown below:
C L= + t = = Σ LISE(X+11,)
+ Ae Σ CrossEntrops(0 t
(11)
iEG.
where X1+1. is the ground truth observation, and E indicates the existence of an edge in the causal graph G.. The importance weights Xy = A = 1 when training the full model, but X₂ is set 0 when learning without the physics-awan aware causal structure while xy is set to 0 when learning the causal structure alone.
Non-causal labels for physics-aware causal structure learning. In Section 3 we assume that the causal relations are given explicit labels based on the prior equation (Eq. 5 The PDE provides information regarding which past and neighboring variables can be considered as possible causes for current variable. It does not, however, provide information about which causal relations should be excluded. Since the physics-aware causal labels are highly imbalanced, CM will overfit on the positive-only labels. We address this challenge by introducing non-cate non-causal labels based on the principle that an effect can not occur before its cause. The non-causal labels are described as {n timestamp (t,) of candidate cause (Xt,,) is later than that of a candidate effect (X,,.). Despite the availability of the non-causal labels, the imbalance issuc still exists as the cardinality of (n f5t) is much larger than that of "p We mitigate this by subsampling the non-causal labels as many times as the available causal labels. More details about model configurations and training settings can be found in Appendix B
4 - tj < 0}, capturing the set of relations where
5 Experimental results
We evaluate the proposed method in terms of both cansal structure lcarning performance and dynamic forecasting performance. For causal structure learning, we evaluate the causal module (STGC) using synthetic and benchmark time series data. We further evaluate the performance with hidden confounding. For dynamic forecasting performance, we evaluate P-STGGN through a graph signal prediction task with real-world observations. Lastly, we discuss the scenarios where only partial prior equations are accessible.
5.1 Causal structure learning evaluation
Given N different stationary series (or nodes), we train a model to predict if there exists significant temporal causal relationships between two time series: X and X,,- Since the auto-regressive order is P, there
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are potentially NP x N causal relations from N variables X where t' E [t - P,t 1 to N variables at time 1. The true temporal causal relations are explicitly given as labels during training and model is evaluated in two different aspects: (1) inter-causality classification and (2) intra-causality retrieval. For the inter-cousality classification, we split a simulated multivariate time series into two parts across time axis: [X,]== = ;
] ;Terrn) and {X,] = Terain:".... ; For the intru-causalily relrieval, we only use a subset of the known labels to train a model and evaluate can retrieve the unseen labels correctly.
Baselines The task can be considered learning directional edge representations from a variable at t' f Pt- P,t 1] to another variable at t, inspiring the three baselines follows. First, we feed two node values into an MLP to predict the strength of causality. The other two basclines utilize a spatial and a temporal module to aggregate neighboring spatial/temporal values accordingly, after which the aggregated two node features are fed into an MLP to return the causal probability. For the spatial encoder (SE), we use GCN (Kipf & Welling, 2017). Chebyshev graph convolution networks (CHEB) (Defferrard et al., 2016), and GRAPHSAGE (Hamilton et al.] 2017). The temporal encoder (TE) then concatenates node variables in the ruto-segressive order. The STGC combines the two encoders spatiolemporally and the resultant node representations are fed into an MLP. Furthermore,
compare to causal discovery baselines: PARC (PCMCI (Runge et al. (019bb) based on partial correlations), Gaussian process regression and a distance correlation (GPDC), DYNOTEARS (Pamfil et al. 2020 and the SOTA Amortized Causal Discovery (ACD) (Löwe et al. 2022
5.1.1 Synthetic study
We first generate multivariate time series X E R from known temporal causal relations. Consider N different stationary time series where each series influences the others in a time-lagged manner. At time t, a variable in i-th scries X R defined as a function of variables at t' < such that di = E Ex=1 f (Xrs) + described in (Runge et al. 2019b). P is the auto-regressive order across time and is noise term that is independent of other variable. Note that (+) is regarded as causal function from previous variable at (j,4") to current variable (i, Since the time series are stationary, the function (+) can be relaxed as f/ (.). We defined the temporal causal function in two different ways: (1) linear, and (2) non-linear conditional independence. For both settings, we generate length T 1000 time series across N 7 (linear) and N 13 mon--ineer)) nodes. More details can be found in Appendix C
Inter-causality classification The results clean data (provided in Appendix C demonstrate that the proposed model successfully oulperforms other baselines on both settings. To further evaluate the robustness of the proposed model, we intentionally add i.i.d. noises to the generated time series. Since the time series are "contaminated" by the random noise after being cansally genorated, it becomes much more difficult to discover underlying temporal causality. Table [ shows AUC of the models on the linear and non-linear settings. While AUCs are commonly decreased compared to the results on clean data, STGC can still learn meaningful representations from the spatiotemporal series unlike other methods. Note that when the scale of noise is increased (O(0, 53)), MLP and spatial encoders followed by MLP are almost impossible to distinguish causal and non-causal relations (AUC is R 0.5), occurring also for ML+ for the non-linear series.
Furthermore, ACD requires a large amount of training data to produce accurate causal classification (its default training size is 10,000). In Table D the performance of ACD suffers from the limited training data; additive noise is generally a significant bottleneck for existing temporal causal discovery methods in multivariate time series scttings. STGC, however, can learn robust representations for effective causal discovery. Supportive results are discussed in the Appendix C comparing STGC against PARC, GPDC, and DYNOTEARS. Through comparisons, STGC is shown to be robust in causal classification with noisy and limited data by utilizing physics-aware causality.
Intra-causality retrieval We consider the time series generated from non-linear causality with added noise for evaluation. Note that there are 21 causal relations in the series that are split into two parts for training and testing. By adjusting the number of causal relations shown training scries, we can evaluate the robustness of the proposed model when the majority of causal relations are not given during the training process. Table shows the average behavior when training on subset of causal relations in time series.
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Table 1: Inter-causality classification with additional noise.
Linear causality (AUC)
Non-linear causality (AUC)
Model
(0 19)
N(0,,5 52)
N(0,13)
(N 5²)
MLP
0.611+0.029
0.506+10.010
0.517+000013
0.499:H0.004
GCN+MLP
0.507=0.004
0.5001:.001
0,5024400022
0.500+10.004
CHEB+MLP
0.627=0.010
0.513+0.008
0.526+0.009
0.500=0.004
SAGE+MLP
0.621
0.516+1.006
0.527+0.007
0.502+00.003
TE+MLP
0.827=0.021
0.697#0.012
0.562+0.033
0.511#0.009
ACD
0.476:10.031
0.489#0.020
0.405
0.504
STGC(Ours)
0.84910.020
0.71210.013
0.640:10.012
0.582:10.007
Table 2: Intra-causality retrieval (AUC) from non-linear causal time series with /o 12)
Number of train causality labels
Model
16 /
11 / 10
6 / 15
1 / 20
ME+
0.530=+0.031
0.546
0.539+00.028
0.501=0.011
ACD
0.486=100030
0.506:10.014
0.499:1.0012
0.500=0.016
STGC (Ours)
0.636:10.024
0.620:0.010
0.596=0.014
0.585=0.018
While TE+MLP delects some unseen causal relations when the number labels shown for training is large (16), its performance quickly degrades as the number of available labels decreased. STGC outperforms TE+MLP by a large margin, supporting the claim that STGC can extract more informative spatiotemporal representations. Interestingly, even if only single causal relation is given as known label (1/20), STGC still manages to retrieve unseen causal relations. Due to the lack of sufficient training samples, the ACD fails to perform effective causal discovery. Compared to ACD, STGC, by leveraging the physics-aware causality, is able to retrieve unseen causal relations.
5.1.2 Evaluation on benchmark datasets
Three benchmark datasets considered in the literature (e.g., Löwe et al. (2022) are employed: Particles, Kuramoto [Kurmooo 1975). and Netsim (Smith et al. 2011). Particles and Kuramoto are two fully-observed physics simulations. The Particles dataset simulates five moving particles in 2D, and some particles can influence others by pulling a spring. The Kuramoto dataset simulates five 1D time-series of phase-coupled oscillators. For both datasets, we follow the same settings as the synthetic study and generate T 1000 time series for training. The Netsim dataset contains simulated fMRI data, and connectivity is defined between 15 regions of the brain. We follow the same settings as reported in (Löwe across 50 samples.
2022 and infer the connectivity
Table B shows the results comparing STGC to ACD the three benchmark datasets. From the results, we see that STGC significantly outperforms ACD. For example, on Kuramoto, P-STCGN achieves 40.6% accuracy improvement compared to ACD. On Particles, the performance of ACD is particularly poor due to the reduced data size. Though ACD achieves 0.999 AUC on the Particle dataset with sufficient data (10,000) as reported in (Löwe et al.) 2022), its performance drops significantly with limited data (1,000). In contrast, we show that P-STCGN is capable of learning robust representations that aid in settings with limited data.
5.1.3 Hidden confounding
In this experiment, we study the effectiveness of the proposed approach with hidden confounding. We use the Particles datasot, where variation introduced through an unobserved temperature variable. The temperature serves modulate the strength of interactions among particles, where higher temperatures correspond stronger forces and consequently, more chaotic system. For each individual sample in the dataset, we
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confcurding
0.4
Benchmark dataset.
Model
Particles
Kuramoto
NetSim
0.2
ACD
0.493
0.562
0.688
STGC (Ours)
0.520
0.968
0.925
002
Table 3: Comparison to ACD on benchmark
datasets.
Figure 3: Forecasting with Hidden Confounding (MSE)). Shaded regions represent the standard deviation across trials.
Table 4: Summary of results of prediction error (MSE) with standard deviations.
TMAX
TMIN
Model
Western
Eastern
Western
Eastern
DCRNN
0.1824
0.1585++0.00 0033
0.0707=0.0017
0.1317=0.0029
GCRN
0.1336+000082
0.1588++0.002
0.0701+0.0004
0.1302=0.0009
FNO
0.1234+0.0005
0.1963 0003
0.090610.0004
0.16761 0.0001
PA-DGN
0.2620#000033
0.2921+:0.0014
0.1720=0.0098
0.2346# 0.0009
P-STCGN
0.111140.0014 0.1355++0.0034
0.0701+0.0009
0. n.1262-0.0000
SNOW
PRCP
Model
Western
Eastern
Western
Eastern
DORNN 0.0057
0.0406::00000 00002
1.4703+000000
0.758810.0013
GCRN 0.6083::0.0022
0.0406++000001
0.4703+100009
0.7595=0000011
ENO
PA-DGN
0.662 0.0402+0.0027
0.4979++0.0016
0.6819=0.0008
P-STCGN
0.6613+0.0035 0.0386:0.0007
0.4589+0.0033 0.6658=0.0025
draw an independent lemperature T value from categorical distribution T Cateeorical([a/22, 2al) following Löwe et al. (2022). This categorical distribution provides a probabilistic framework for assigning different temperature levels, allowing for the simulation of diverse scenarios within the particle system. In total, we consider four settings with categorical parameters a = 10.1,
To model the temperature variable as hidden confounder, we introduce hidden node in our model that affects the status of all particles. We also consider baseline where hidden confounder is assumed. We report both the structure learning accuracy (AUC) (Appendix Fig. 5] and the forecasting performance (MSE) (Fig. 3 As shown, including the hidden confounder in our model effectively achieves better causal structure discovery performance with higher AUC across different settings. In terms of forecasting performance, being able to model hidden confounders also achieves improved forecasting performance, especially with lower temperatures (i.e., a = 1,2).
5.2 Dynamic forecasting evaluation
To evaluate the dynamic forocasting performance of the proposed model, we consider a graph signal prediction task from real-world observations. The task a prediction of future signals X given length 10 past spatiotemporal series X,-9... X, under the graph structure.
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Table 5: Data efficiency evaluation. 60% represents the full training set.
% of available data used
Measurement
Model
5%
10%
20%
60%
STGF
0.1926-+0.0937
0.1228
0.1177+0.0029
0.1134++0.0003
TMAX (West)
P-STCGN
0.138211.0034 0.120410.0005
0.1186=0.0007
0.1111#0.0014
STGF
0.1611=0.0047
0.1519+000039
0.1410=#00039
0. 0.1393=0.0011
TMAX (East)
P-STCGN
0.1584+0.0043 0.1493=00.0022
0.1404+0.0013
0.1355+0.0034
STGF
0.1229#0.0120 0.096310.0063
0.0887=000000
0.0759=0.0024
TMIN (West)
P-STCGN 0.1059+0.0080
0.0976++00012
0.0074+0.0014
0.0731+0.0009
STGF
0.1571=000000
0.1352
0.126310.0116
0.130410.0033
TMIN (East)
P-STCGN 0.1427+100004
0.1283+10.00%
111332+0.003
0.1262+0000066
STGF
1.3300+000685 0.9087+000100
0.8150
0.6720+000070
SNOW (West)
P-STCGN
1.222310.0051 0.9083
0.797710.0051
0..003
STGF
0.046010.0014 0.0010
0.0362+100000
0.0090
SNOW (East)
P-STCGN
0.043910.0005 0.0412+0.0083
0.0356+0.0001
0.0386=0.0007
STGF
0.5103:1000022 0.4020:00.0000
0.4407
0.46194000047
PRCP (West)
P-STCGN
0.508410.0012 0.4627:0.0014
0.4437#0.0022
0.4589=0.0033
STGF
0.80281000029 0.8041+000000
0.7980=0.0051
0.6770:000042
PRCP (Fast)
P-STCGN
0.7982+0.0029
0.7981+0.0029
0.7884+0.0012
0.6658 0.6658+0.0022
Dataset We consider the climatology network Deeferrard et al. 2020). Each sensor has different daily measurements: TMAX: Maximum temperature (tenths of degrees C), TMIN: Minimum temperature (tenths of degrees C), SNOW: Snowfall (mm), and PRCP: Precipitation (tenths of mm). Each measurement is provided over years from 2010 to 2014 (the length of series 1826), and we use them for our experiments. It. is worth noting that the number of working sensors for each measurement highly variable. While daily temperature obscrvations are spatially densely available, the snowfall obscrvations are comparatively sparsc. Table 12 provides additional details for the dataset. The number of sensors from which the underlying graph constructed is listed (along with the number of edges in the resulting graph). We split the series into training (60%), validation (10%), and testing (30%) sets. Additional details are in Appendix D
Baselines We compare P-STGGN against two well-established data-driven baselines which have been introduced for similar tasks: DCRNN (Li et al. 2018) and GCRN (Soo et al. 2018). For physics- based baseline, we consider the PA-DGN (Seo et al. 2019 and FNO (Li et al. 2020). To adapt FNO for forecasting, we train it on observational data using its default settings and test it for a future time step prediction.
Causality labels from PDEs There no ground truth PDEs for this dataset. We thus consider the PDEs among the family of the continuity equation, e.g., Navier-Stokes equations. These equations commonly describe how target observations are spatiotemporally varying with respect to its second-order spatial derivatives and first-order time derivative. In the underlying graph structure, spatially 1-hop neighboring nodes (j EN) are considered as adjacent causes to the obscrvation at the 1th node, and observations at are palential causes to the observations at. autoregressively. The existing causal labels can be described as f - and j Mi}.
5.2.1 Prediction accuracy
We use mean squared error (MSE) as a metric to compare P-STCGN against the external baselines 2 Table shows that P-STCGN mostly outperforms other baselines across different regions and measurements. Both
Clobal Historical Climatology Network (CHCN) provided by National Oceanic and Atmospheric Administration (NOAA). For SNOW and PRCP, FNO unable converge during training, likely due the spatially sparse sensors with discrete
measures.
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Table 6: Forecasting with partially available prior (MSE)
% training data
# equations withheld 100%
10%
3.287
4.71
3.27+0.18
4.72:20.35
3.36:0.16
4.86=0.49
3.30:L0.19
4.99
DCRNN and GCRN replace fully connected layers in the RNN variants (GRU and LSTM) with diffusion convolution and Chebyshev convolution layers. Thus, they similarly aggregate spatiotemporal correlation, exemplified by the close prediction error. Compared to the data-driven DCRNN and GCRN, our approach achieves better accuracy, particularly for TMAX and PRCP. For example, P-STCGN improves DCRNN by 16% for TMAX (western) prediction. Compared to two physics-based basclinos PA-DGN and FNO, we also observe performance improvements. Particularly, P-STCGN improves PA-DGN significantly for TMAX and TMIN, demonstrating that our means of incorporating prior physics knowledge is much more effective.
5.2.2 Ablation study
To further study the effectiveness of P-STCGN, we perform ablation studies on data efficiency and gener- alization ability. We compare its performance to the baseline model STGF (i.e., just the forecasting base without explicitly incorporating physics-informet informed causal signal).
Data efficiency We consider different training sets with reduced number of training samples of 5%, 10,, and 20%. In comparison, we also consider the full training set (60%). Results are shown in Table 5 Compared to STGF, we can clearly see how the additional physics-aware causality is beneficial for modeling spatiotemporal data, particularly on extremely limited data (5%). Specifically, P-STCGN improves STGF by 28% on TMAX (western), implying that the PDE- based causal labels are significantly informative and can help compensate for the lack of training samples. As the training set size increases, the MSE difference between P-STCGN and STGF reduces. Nevertheless, P-STCGN continues to outperform STGF by decent margin.
Generalization Ability We also consider the generalization ability, and a proxy we train P-STCGN and STGF on one region and test on another. We observe that P-STCGN consistently outperforms STGF in most scenarios. Detailed results are reported in Appendix D
5.2.3 Interpretation of learned causality
Once the causal module is trained based on a prior PDE, we can use it to examine how potential causes cary over space and time. We visualize how the causal probability is changed in Appendix D From the visualizations, we can observe that variables spatially close to current obscrvations have higher causal association for PRCP and TMAX, Additionally, we can infer the strength of causal relations between neighboring sensors and a specified target sensor (as shown in Appendix D) As shown, the physics-aware causality is not only informative for spatiotemporal modeling but also enables the discovery of unspecified causal relations.
5.3 Discussions on partially available prior
We investigate the offectiveness of utilizing partially available physics priors to enhance forccasting performance, addressing a common challenge in real-world applications where complete prior knowledge is unavailable. Employing the synthetic experiment settings detailed in Section 5.1.1, we utilize synthetic data for forecasting ovaluation, focusing on the linear case with 7 equations. To evaluate, we present forccasting performance under different scenarios, varying the number of equations withheld. For comparison, we report performance with both 100% training data and only 10% training data. The results summarized in Table 6 The
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observed trends indicate that the presence of physics prior, even when partially available, contributes to improved forecasting accuracy, particularly in low-data settings.
6 Conclusion
In this paper, we introduced a novel physics-aware spatiotemporal causal graph network (P-STCGN). This approach enables dynamic forecasting alongside a learned causal structure, capturing relevant information from physics domain knowledge. We evaluated the proposed framework from two primary perspectives: inter-causality classification and intra-causality retrieval. Our experimental results highlight the effectiveness of the physics-aware causal learning approach compared to common alternatives, especially in scenarios involving noisy and limited data. We further evaluated the forecasting performance real-world observations from climate systems, and we observed superior accuracy due to our method utilizing physics-informn causal relations, even when the physics priors are inaccurate.
Acknowledgements
This work was supported in part by the National Science Foundation under awards #2226087 and # #1837131, in addition to the USC Ershaghi Center for Energy Transition (E-CET).
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A Physics-aware Causality
Assume that the physics equations in E satisfy the properties of linearity, causality preservation and the absence of instantaneous feedback loops. Define the construction of the causal graph G. (V,E) based on the physics equations. Let each variable correspond to node, and each directed edge represent a causal relationship. In linear systems, each variable's rate of change is a linear combination of its influencing variables. According to the causality preservation, each variable's rale of change is solely determined by the values of variables its causal neighborhood. Hence, there are directional dependencies from influencing variables to influenced variables. Both lincarity and causality prescrvation prevents circular dependencies, ensuring that there are no loops formed among variables through the causal relationships derived based physics equations. No instantaneous feedback property further eliminates the possibility of conclude
Proof of Theorem 1:
loops involving instantaneous feedback in the causal graph Ge. Combining three properties, that each variable's rate of change is dictated by linear combination of its immediate causes, and there are no circular dependencies introduced. Therefore, the derived causal graph is acyelic.
B Model configuration and experimental details
B.1 Model configuration
STG consists of two parts: SE and TE, and STG shared in STGC and STGF. SE is defined by two-layer of GraphSAGE with 32 hidden units. TE concatenates the output of SE in a temporal axis, [X,-p; X.]. CM an MLP [FC32,ReLU,FF] Siggmood,, and VM is an MLP FcLU, ReL,U where FC(n) denotes a fully-connected layer with n units. The baselines are defined to have a similar number of learnable parameters to P-STCGN.
B.2 Training settings
We train P-STCGN for all tasks with batch size of 32 on single GPU (NVIDIA T4 GPU) for 1000 epochs with early stopping (ending early when the validation error is not improved for 20 epochs). All results in the paper are mean values from 10 different random seeds.
In Table 7 we report training times for both STGF (no causal module) and P-STCGN (with causal module) on the dynamic forecasting task using the climatology network presented in Section 5.2 Reported times are averaged across 10 trials the above mentioned hardware for 1000 training epochs over the respective time series (TMAX, TMIN, SNOW, PRCP). Note that the number of available samples for each series varies; see Table 12 for additional dataset details. Inference times were also measured post-training, averaging to 3.5s per 100k output values across all settings. Inference times are approximately uniform given the underlying model architecture is the same for each setting. Table reports the number of parameters in each of the model components used for this task. Taken logether, these tables provide a cohesive view of the datasel size, model size, and computational cost required to reach the reported performance, serving as a loose guide for understanding the scale of model deployment on other real-world tasks.
C Causal structure learning evaluation
C.1 Synthetic time series data generation
Data generation We first generate multivariate time scries X E T from known temporal causal relations. Consider N different stationary time series where each series influences the others in a time-lagged manner. At time f, a variable in the i-th time series Xx, E R is defined as function of variables at t' < such that:
(-1
Fi. =
(12)
1'=
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Table Observed training times over the 1,000 epoch horizons, averaged across the trials reported for the climatoloyy task in the main paper.
TMAX
TMIN
Model
Western
Eastern
Western
Eastern
STGF
5974.93s=117.009
3520. 188
6124.94s:t88.15
3517.356=79.61s
P-STCGN
6373.39s
3872.72x-441.845
7247.38%+99.886
3919.36s+110.299
SNOW
PRCP
Model
Western
Eastern
Western
Eastern
STGF
678.63s 6...63$111644 1.64s
1711 7711.299+3..44
7710.77s+96.44
4678.70s+144.299
P-STCGN
764.53s 1.36s
6.788
54.997
1942.195-15.977
Table 8: Parameter counts for model components used on the climatology forecasting task.
Model component
of parameters
Encoders (spatial/temporal)
2,176
Causal module
22,593
Value module
12,385
STGF P-STCGN
14,561
37, 154
described in Runge et al. 2019b Further breaking down this formulation seen in Section 4.1, we generate two scries based on lincar and nonlinear causality:
Linear causality
X1, 0.7X,-1,0 +
X₂, 0.8X,-1.1 0.8
X., 0.5X, - 0.5X,-2,1 0.6X,
X,3 0.4X,
X,, 0.9X, 2-2 0.1 +
X,,5 = 0.2X + 0.2X, + 0.2X, +c
X =E
where e <~ (0, 1). the linear causal series, there are 12 causal relations between = 7 series and the maximum time lag in the causal relations 3.
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2.5
MA Hel
2.5
0.0
10:
28
oo phart Ms
-5
10
-25
sertee
2.5
0.0
2.5
-10
0.0
2.5
-2.5
0.0
-2.5
3rdd
10
-10
70
-20
6ith ceriee
10:
2.5
deriee
0.0
0.0
2.5
2.5
-2.5
12th
0.0
0.0
-2.5
-2.5
12th
400
800
1000
200
400
600
800
1000
(a) Generated multivariate time series
(b)
Generated multivariate time series
from given linear causal relations,
from given nonlinear causal relations.
Nonlinear causality
X1,0
X,, 0,2( +.7X, +
X1, = 0.3(X,-20) -,0
X,,3 = -0.09 + -,,
X,,4 0.2(X. 0.01 0.2( 0.2 +e
XG,5 =
X,,6 = 0.3X 1-5 + 0.3X, -,3
X = -0.2(X 0.7X;
X,, (0.((x--1))) (x,()x)
X1,9 = 0.9X +c
X = -0.22X 1-0 0.1X
-3,, 0.2(X, +c
X,1I = -0.3X 4,3
X = -0.3X,
where (0, 1). In the nonlinear causal series, there are 22 causal relations between N = 13 series and the maximum time lag in the causal relations is 4 (See X,,1). When we conduct the intra-causality retricval experiment, we feed length 4 series from X to X, to the classifier. Thus, the causality from time lag in X,,1 is not labelled. Figure 4b shows generated sample time series based on the formulation above.
Data preprocessing Given N different generated time series (length T 1000), we assume fully connected graph structure between the multivariate time series. We the first. 50% of the series data for
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Table Inter-causality classification (clean data)
Linear causality
Model
Recall
AUC
СE
MLP
0.579-
0.670=0.012
0.611
GCN+MLP
0.193-10112
0.508=0.008
0.669=0.004
CHEB+MMP
0.577=0.055
0.677=0.010
0.585=0.017
SAGE+MLP
0.554
0.668=0.035
0.583
TE+MLP
0.756=0.038
0.858=0.020
0.435=0.026
STGC
0.767=0.023
0.885=0.011 0.340=0.035
Non-Linear causality
Model
Recall
AUC
CE
MLP
0.365-10.211
0.533=0.023
0.658
GCN+MLP
0.241=0.194
0.511=0.002
0.677=0.013
CHEB+MLP
0.416=0.124 0.551=0.013
0.650=1.001
SAGE+MLP
0.367=0.101 0.554=0.006
0.637=0.015
TE+MLP
0.418
0.611=0.051
0.625+0.017
STGC
0.503-0.041
0.689=0.013
0.522+0.015
the training series, and the following 20% for the validation series. The remaining 30% of the series is used evaluate the baselines and P-STCGN.
C.2 Ablation study on clean synthetic data
For the inter-causality classification task, we report the mean of recall, AUC, and cross entropy crror (with standard deviation) on the test series. In this task, the temporal causality among the potential relations (NP N) sparse, implying the recall, which tells how many actual causal relations are retrieved, is particularly important. We evaluate the proposed model on two different settings: (1) linear, and (2) non-lineat temporal causality. The results in Table demonstrate that the proposed model successfully outperforms other baselines on both settings. More specifically, all models are able to distinguish non-causal and causal relations in the linear setting according to AUC. However, the temporal change is particularly important to understand the causality among the variables. For the non-linear selting, the results show that metrics from models are degraded significantly compared to the linear setting. Nonetheless, the temporal information more important but the spatial information can still be helpful (STGC vs. TE+MLP).
C.3 Comparison to causal discovery methods
Table shows recall from existing causal discovery in multivariate time series methods (PCMCI based on partial correlations (PARC) and Gaussian process regression and a distance correlation (GPDC)) on the non-linear series. It shows that STGC is able to learn robust representations for the causal discovery from noisy series by utilizing the explicitly given labels.
Forecasting evaluation
D.1 Real-world graph signal data
Data preprocessing We first sample time series from the entire sensor array to construct more localized graph signals. The number of available sensors dependent on the type measurement. Given multivariate time series from multiple sensors, we construct distance-based graph structure using a k-NN algorithm where k 2. The value of is chosen such that graph density is properly balanced, and to ensure that
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Published in Transactions on Machine Learning Research (07/2025)
Table 10: Generalization evaluation across the two regions.
TMAX
Western
Eastern
Model
Train on Western
Train on Eastern
Troin on Eastern
Train on Western
STGF
0.113410.0014
0.125610.0027
0.1393=0.0011
0.155620.0024
P-STCGN 0.111110.0014
0.1240:0.0015
0.1355:1.0034
0.1532::0.0019
TMIN
Western
Eastern
Model
Train on Western
Train on Eastern
Troin on Eastern
Train on Western
STGF
0.0759:0.0024
0.090610.0040
0.1304+0.0038
0.1308::.0028
P-STCGN
0.0731+0.0009
0.0919+0.0021
0.1262+0.0036
0.1284+0.0014
Table 11: Recall for causal discovery methods
Noise
PARC
GPDC
DYNOTEARS
STGC
(o,, 12)
0.48
0.48
0.29
0.66
52)
0.00
0.00
0.00
0.48
sensor is only connected with other spatially close sensors. It worth noting that the number of working sensors for each measurement highly variable. While daily temperature observations are spatially densely available, the snowfall observations are comparatively sparse. Table I provides additional details for the dataset. The number of sensors from which the underlying graph was constructed listed (along with the number of edges in the resulting graph).
Each observation is multiplied by a scalar (0.01) to be normalized and provide numerically stable computation. We used the first 60% of the series data for the training set, and the following 10% for the validation series. The remaining 30% series is used to evaluate the baselines and P-STCGN.
D.2 Ablation ability on graph signal prediction
Generalization ability. To further study the effectiveness of incorporating the physics-aware causality, we study the generalization ability P-STCGN. In particular, we perform an ablation study where we train P-STCGN and STGF on one region and test on another region. We consider the TMAX and TMTN for ovaluation and the results are reported in Table 10
D.3 Interpretation of learned causality
Once the causal module is trained based on a guiding PDE, we can use the module to examine how the potential causes are varying over space and time. In the following, we show how the causal probability is changed the two regions. P-STCGN extracts causality-associater information from spatiotemporal series. In Fig. Ga PRCP and TMAX. A similar pattern appears on another region (shown in Fig. 6b) On the other hand, sensors for SNOW are more related to sensors farther away.
see that variables spatially close to current observations have higher causal assocation for
We can also infer which neighboring sensors have stronger/went /waakr causal relations to specified target sensor. In Fig. (6c and [d] 4 sensors are sampled from cach region to visualize how much their K K-hop neighboring variables are causally related. We can see that daily max temperatures from the sensor 42 in the western region have been stroggyy affected by spatially close (2-hop) sensors, however, the max temperature at the
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Published in Transactions on Machine Learning Research (07/2025)
Table 12: Information on sensor networks from the climate dataset.
Western
TMAX
TMIN
SNOW
PRCP
of sensors
434
423
31
319
# of edges
1142
1110
76
862
Eastern
TMAX
TMIN
SNOW
PRCP
# of sensors
244
248
114
323
# of edges
632
636
298
844
Structure Learning Performance
AUC (tigher is Better)
confounding W confounding
0.54
100
155
2.0
255
3.0
35
4.0
Figure 5: Causal discovery with Hidden Confounding (AUCf). Shaded regions represent standard deviations.
sensor 25 is more likely dependent on sensors a bit far away (6 or 7-hop). On the other hand, sensor 26 is more dependent on mid-range sensors (4 or 5-hop). In eastern states, sensors and are associated with closer sensors; however, sensor and do not have distinct causal relations from their neighboring sensors. We find that the physics-aware causality is not only informative for spatiotemporal modeling directly but also enables the discovery of unspecified causal relations.
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Published in Transactions on Machine Learning Research (07 (07/2025)
TMAX (Westem)
SNOW (Western)
TMAX (Eastem))
SNOW (Eastern) PRCP (Eastem)
TMIN (Westem)
PRCP (Westem)
TMIN (Eastem)
Causal probabllity
Causal probability
# of hops
of hops
(a) Average causal probability curves vs. the number of hops over all sensors in Western region.
(b) Average causal probability curves vs. the number of hops over all sensors in Eastern region.
sensor sensor 36 sensor 26
sensor sensor sensor
sensor
sensor 42
Causality in TMAX
Causality in
Causa probability
Causal probability
0
# hops
of hops
(c) Average causal probability curves vs. the number of hops from particular sen- sors in Western region (TMAXX.
(d) Average causal probability curves vs. the number of hops from particular sen- sors Eastern region (TMAX).
22


Reviews:
[{"review_id": "sjdz0QWVD0", "reviewer": "Reviewer_kcCi", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n\nNovel Architecture: The progressive fusion of spatio-temporal and channel-wise representations is original and interesting.\nChannel-wise Graph Learning: Adaptive channel correlation modeling is less explored and adds depth to the understanding of motion patterns.\nStrong Performance: The model achieves competitive results on benchmark datasets, reinforcing its effectiveness.\nDetailed Ablation Studies: The impact of each module is well-analyzed and visualized, showing clear contributions from design choices.\nClarity in Motivation: The paper clearly justifies why modeling both spatial and channel relationships can benefit skeleton-based action recognition.\n\nWeaknesses:\n\nInsufficient Comparison: Missing comparison with very recent baselines like DSTA-GCN or SGN++. Would help position P-STCGN more clearly.\nEfficiency Metrics Missing: No clear discussion on computational cost, parameters, or inference time, important for deployment scenarios.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes P-STCGN, a novel Progressive Spatio-Temporal Channel Graph Neural Network designed to improve human action recognition from skeleton data. The architecture integrates temporal attention, progressive residual fusion, and an adaptive graph learning module that selectively models joint-wise and channel-wise correlations. The model is evaluated on benchmark datasets, showing state-of-the-art or competitive performance. The core contribution is the combination of fine-grained channel correlation learning and spatio-temporal joint modeling in a progressive, attention-driven fashion", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Compare with Recent Models: Include recent baselines for a fairer performance landscape (even just as discussion about differences and justification of why not included as proper baseline)\nEfficiency Discussion: Add a paragraph on computational efficiency.\nThis work makes a strong technical contribution and presents a novel way to combine spatial, temporal, and channel-wise features. While performance is strong, the paper would benefit from improved clarity and better context among the latest baselines. With some polishing and added reproducibility, it has the potential to be impactful.", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "qsbcOETego", "reviewer": "Reviewer_R7Ye", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths:**\n1. This paper proposes an innovative framework that incorporates physical laws and causal analysis. This framework overcome the inexplainability of pure data-driven methods. Besides, this framework ensures alignment with physical rules when data collected is not informative enough.\n2. The experiments show that the method proposed is robust to limited data size. With only 5% training data, the P-STCGN significantly outperforms the baseline method under different metrics.\n3. The method proposed is robust to noisy or biased data. The extensive experiments conducted on synthesized or semi-synthesized datasets show that with additive noise the method still performs well and the inclusion of hidden confounder can mitigate the bias in data.\n4. Theorem 1 in this paper shows that physical formulas satisfying specific regularity conditions can induce DAG, thus theoretically justifying the introduction of physical formulas.\n\n**Weakness:**\n1. As the synthetic experiment shows, incomplete physical equations lead to degraded performance, which means that this method relies on the correctness of physical rules given.\n2. The method proposed includes a causal module with computational complexity of $N^2 K^2$, and the cost relies on the stationary assumption to be reduced, which limits the application of P-STCGN to non-stationary systems.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes P-STCGN, a spatiotemporal causal graph network incorporating physical rules as priors. The model is a novel approach to integrating universal physical equations with data-driven causal structure learning, thus benefiting spatiotemporal forecasting under limited data. The contributions made in this paper include both methodological and applied aspects, which can be summarized as follows:\n1. The proposed method for incorporating information from physical equations as a prior for causal structure learning. This idea is novel and addresses the limitation of data-driven causal discovery algorithms that ignore actual physical meaning and lack generalizability.\n2. This paper designs the P-STCGN model, in which the STGC module is able to efficiently utilize physical equations such as PDEs for semi-supervised causal structure learning, while the STGF is able to utilize the learned causal associations to benefit spatio-temporal sequence prediction.\n3. The extensive experiments conducted on synthetic data, benchmark data, and real-world climate data show the effectiveness of the proposed P-STCGN, even with limited training samples.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "I don't think any critical changes are required.\n\n**Non-critical recommendation to strengthen the work:**\n1. Benchmarking runtime and memory usage during training/testing against baselines.\n2. Showing how the learned causal probabilities correlate with the physical variables or parameters.\n3. Visualizing the learned causal structure and comparing it with the physically guided one. For example, whether the method learns the 13 causal relations in Fig. 1.", "broader_impact_concerns": "I have no concern about the ethical implications of this paper.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "OnlXrsYaj6", "reviewer": "Reviewer_q6C5", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- The paper successfully bridges physics-based inductive biases and causal inference in spatiotemporal forecasting. The use of domain-specific PDEs as causal priors is both innovative and broadly relevant across various scientific and engineering domains.\n- The research addresses a well-motivated and practical challenge, demonstrating clear advantages when forecasting with limited and noisy data—a common scenario in real-world applications.\n- Extensive experimental evaluations validate the effectiveness and robustness of the proposed method.\n\nWeaknesses:\n- The presentation of the paper requires improvement for better clarity and accessibility. Specific suggestions for enhancements are provided below.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper proposes a novel Physics-aware Spatiotemporal Causal Graph Network (P-STCGN) designed for forecasting spatiotemporal dynamics, especially effective in scenarios with limited and noisy data. Its main contributions include:\n- Introducing a method to derive causal structures from spatiotemporal data using domain-specific physics knowledge.\n- Formulating a specialized neural module to learn these causal structures.\n- Providing a forecasting module operating under these causal structures, enabling predictions guided by physics-informed causality.\n- Extensive experimental validation demonstrating robustness and superior performance against several baselines across synthetic datasets and real-world climate data.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. Could you please elaborate clearly on the difference between the graphs $G_s$ and $G_c$?\n2. On page 3, in the bottom paragraph, could you elaborate on why the Heat equation (Eq. 1) elucidates exactly $N_c=13$ causal relations (5 temporal and 8 spatial)? A clear breakdown would help readers better understand your point.\n3. Please ensure notation consistency. In Fig. 2, you denote the Spatial Encoder and Temporal Encoder with \"GConv\" and \"FC,\" respectively, while in section 4.2, you refer to them as \"SE\" and \"TE.\" This inconsistency may cause confusion and should be clarified.\n4. In section 4.1, could you define nodes $V$ more explicitly? Earlier in the paper, each node corresponds to a single variable, but here each spatial location is defined as a node containing multiple variables (e.g., velocity vector and pressure). Clarifying precisely what constitutes a node would improve readability.\n5. Please include a detailed proof for Theorem 1 to support the theoretical claims clearly.\n6. In section 4.2, it would greatly help to explicitly state the training objective (loss function) of your proposed method.\n7. In Theorem 1 (b), \"Causality preservation,\" there seems to be a contradiction: initially, it states, \"each variable's rate of change is solely determined by variables in its causal neighborhood,\" yet the subsequent equation includes an external term $g_i(t)$. Please clarify or reconcile this inconsistency.\n8. In section 5.1.3, you use the notation $t$ for temperature, while earlier you use $t$ for time, causing potential confusion. Please adopt a distinct variable to represent temperature explicitly.\n9. Typos to fix: \n- page 4 (bottom paragraph): \"node $i$\" should be \"node $(i,j)$\". \n- Equation 4: \"$u_{i,i}$\" should be \"$u_{i,j}$\".\n- Page 12: \"$40\\\\%$\" should be \"$60\\\\%$\".\n\nIf the authors address all of these requested changes thoroughly, I would be pleased to recommend this paper for acceptance.", "broader_impact_concerns": "The paper raises no broader impact concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)