Federated Learning of Nonlinear Temporal Dynamics with Graph Attention-based Cross-Client Interpretability

Ayse Tursucular; Ayush Mohanty; Nazal Mohamed; Nagi Gebraeel

cs.LGFeb 13, 2026

Federated Learning of Nonlinear Temporal Dynamics with Graph Attention-based Cross-Client Interpretability

Authors: Ayse Tursucular, Ayush Mohanty, Nazal Mohamed, Nagi Gebraeel

Organizations: Georgia Institute of Technology Atlanta, GA, USA

Abstract

Networks of modern industrial systems are increasingly monitored by distributed sensors, where each system comprises multiple subsystems that generate high-dimensional time-series. These subsystems are often interdependent, making it important to understand how temporal patterns observed at one subsystem relate to those observed across others. This problem is particularly challenging in decentralized settings where raw measurements cannot be shared, and client observations are heterogeneous. Moreover, in practice, each subsystem (client) operates a fixed, proprietary model that cannot be modified or retrained, limiting the applicability of existing approaches. Separately, nonlinear dynamics make cross-client temporal interdependencies difficult to characterize, as they are embedded implicitly within nonlinear state-transition functions. This paper presents a vertical federated learning framework for learning one-step-ahead cross-client temporal interdependencies under these constraints. Each client maps high-dimensional local observations to low-dimensional latent states using a nonlinear state-space model, while a Graph Attention Network (GAT)-based central server learns a state-transition model over the communicated states. Given a candidate graph skeleton, the server learns state-dependent attention coefficients that capture cross-client interactions. We analytically relate the GAT attention coefficients to the Jacobian of the learned state-transition model, yielding a first-order characterization of cross-client interdependencies. We further establish bounded closeness of the learned dynamics and Jacobians to a corresponding centralized oracle. Synthetic experiments validate the theoretical results on controlled systems with known graph skeletons. Experiments on real-world datasets with unknown graph skeletons demonstrate superior federated root cause analysis over baselines.

Figures & tables

Figure 1. Schematic of our proposed framework with communication between client $m$ and the server

Figure 2. Training loss curves on the nonlinear SSM synthetic dataset for clients (left) and server (right) for a M=3 clients.

Method	Linear SSM	Nonlinear SSM	Lorenz-96
Centralized Oracle	0.036	0.005	0.001
Proprietary	0.654	0.117	0.932
Ours	0.101	0.010	0.011
Reduction (%)	84.6	91.4	98.9

Table 1. Prediction accuracy (test MSE) on synthetic datasets.

Figure 3. Average client residuals on the nonlinear SSM.

Method	Linear SSM	Nonlinear SSM	Lorenz-96
FedPC	0.072	0.586	0.499
FedCSL	0.072	0.585	0.499
NOTEARS-ADMM	0.068	0.583	0.497
FDBNL	0.063	0.587	0.500
FedDAG	0.027	0.415	0.477
Ours	0.965	0.956	0.732

Table 2. Cosine similarity: learned vs ground truth Jacobians.

Panel of Figure 4 — Figure 4. Synthetic validation: Correlation between the learned model and the ground truth for (a) attention coefficients and (b) Jacobian operators. (c) Correlation between attention coefficients and Jacobians in the learned model.

	HAI		TEP		SMD
Method	AC@1	AC@2	AC@1	AC@2	AC@1	AC@2
FedPC	0.41	0.76	0.19	0.40	0.41	0.60
FedCSL	0.41	0.65	0.20	0.39	0.43	0.63
NOTEARS-ADMM	0.35	0.76	0.18	0.35	0.42	0.61
FDBNL	0.41	0.71	0.19	0.37	0.44	0.62
FedDAG	0.35	0.65	0.18	0.39	0.41	0.62

Table 3. RCA performance (AC@

k

) on real-world datasets.

Method	HAI	TEP	SMD
Centralized (Oracle)	0.240	0.635	0.377
Proprietary	0.873	0.720	0.712
Ours	0.508	0.699	0.558
Reduction (%)	41.8	2.9	21.6

Table 4. Prediction accuracy (test MSE) on real-world datasets.

Dataset	Corrupted Clients	AC@1	AC@2
HAI	None	0.76	0.94
	1 of 3 (33%)	0.59	0.94
	2 of 3 (67%)	0.59	0.94
	All	0.47	0.82
TEP	None	0.45	0.65
	1 of 5 (20%)	0.40	0.63

Table 5. Robustness under corrupted proprietary models.

Dataset	Server	Client Loss $\downarrow$	Mass on Absent Edges $\downarrow$
Linear SSM	GAT	0.101	0.000
	LSTM	0.101	0.146
	MLP	0.101	0.129
	Transformer	0.101	0.224
Nonlinear SSM	GAT	0.0100	0.000
	LSTM	0.0112	0.457

Table 6. Comparison of server architectures.

Dataset	Server Graph	Client Loss $\downarrow$	Cosine Similarity $\uparrow$
Linear SSM	True Graph	0.1010	0.965
	$\rho=10\%$	0.1010	$0.880\pm 0.036$
	$\rho=20\%$	0.1010	$0.817\pm 0.026$
	$\rho=30\%$	0.1010	$0.721\pm 0.019$
	Fully Connected	0.1010	0.932
	Diagonal Only	0.1000	0.067

Table 7. Graph misspecification study on synthetic datasets.

Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 5. Toy example ( $M=3$ nonlinear SSM, same oracle and server GAT as Section 7 ): edge-by-edge temporal tracking of the Jacobian magnitudes $\|J_{mn}(t)\|$ (standardized), learned server vs. ground truth, over the $3\times 3$ grid of client pairs.

Figure 6. Toy example ( $M=3$ ): edge-by-edge temporal tracking of the attention coefficients $\alpha_{mn}(t)$ (standardized) over the $3\times 3$ grid of client pairs.

Figure 7. Toy example ( $M=3$ ): per-edge temporal correlation between attention coefficients and Jacobian magnitudes.

Figure 8. State dependence of the learned operator (Nonlinear SSM). Left of the dotted line: window-averaged $\|J_{mn}\|$ in an early and a late window, ground truth (top) vs. ours (bottom); each map matches the truth (match cosine annotated). Right: the between-window change map: the learned operator changes in the same places and directions as the truth (change-match annotated). Gray = no edge; color scales are per row.

	HAI		TEP		SMD
Operator	AC@1	AC@2	AC@1	AC@2	AC@1	AC@2
Fixed identity $J$	0.43	0.77	0.20	0.40	0.44	0.64
Fixed all-ones $J$	0.56	0.85	0.40	0.63	0.52	0.70
Ours, learned $J(t)$	0.76	0.94	0.45	0.65	0.53	0.70

Appendix

Table 8. RCA with fixed reference operators in place of the learned

J(t)

, under the same detector, timing, and ranking as Table 3 .

Scale w.r.t.	Value	Server loss $L_{s}$	Read-out (s/step)
	8	$0.0022\pm 0.0001$	$0.0003$
	16	$0.0023\pm 0.0001$	$0.0003$
Observation dim. $d_{m}$	32	$0.0025\pm 0.0002$	$0.0003$
	64	$0.0028\pm 0.0003$	$0.0003$
	128	$0.0031\pm 0.0004$	$0.0003$
	2	$0.0023\pm 0.0001$	$0.0003$

Appendix

Table 9. Scalability: server loss on the test window (mean

\pm

s.d. over 3 seeds; base

M{=}3

p_{m}{=}2

d_{m}{=}16

) and wall-clock time of the Jacobian read-out per time step. The

M

rows keep the cross-client graph density fixed (Table 10 ).

$M$	Cross edges	Possible	Density
3	4	6	$0.667$
4	8	12	$0.667$
8	37	56	$0.661$
16	160	240	$0.667$
32	661	992	$0.666$
64	2688	4032	$0.667$

Appendix

Table 10. Realized adjacency statistics of the

M

sweep: the cross-edge density is

2/3

at every

M

(up to integer rounding of the edge count).

Figure 9. Server loss on the test window versus

\varepsilon

for the two communication directions (one curve per

\delta

), normalized per client and state dimension. Left: client-to-server latent-state perturbation. Right: server-to-client gradient perturbation. Dashed: non-private reference. Shaded bands:

\pm 1

standard deviation over seeds.

Figure 10. Interpretability under privacy on the

M{=}6

nonlinear SSM: cosine between the learned interdependency structure and the ground-truth interaction structure.

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Uncertainty Quantification in Federated Granger Causality Learning

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Georgia Institute of Technology Atlanta, GA 30332

Time-Series Causal Discovery Causal Discovery Federated Graph Learning

Open paper AskPDF arXiv

Granger causality identifies predictive dependencies in multivariate time series. In distributed settings where parties cannot share data, federated causal learning enables joint analysis. Most federated causal methods assume that clients observe the same features and infer causal relationships as point estimates, with little formal uncertainty quantification. These assumptions do not hold in many industrial systems, where clients observe different features, and the objective is to estimate cross-client dependencies (edges). These dependencies must be estimated indirectly through repeated client-server iterations. Uncertainty from client data and model parameters propagates through this process, making point estimates alone insufficient for assessing cross-client edges. This paper characterizes this uncertainty propagation and uses edge-specific variances to distinguish genuine cross-client dependencies from spurious estimated edges. We consider aleatoric uncertainty from client data variability and epistemic uncertainty from model parameters. We derive closed-form variance recursions and steady-state variances for the client-server iterations. We prove that the propagated contribution of the initial model-parameter uncertainty vanishes asymptotically. These variances enable statistically principled selection of cross-client edges. Synthetic experiments show that our approach improves cross-client edge recovery over competing baselines. On real-world industrial datasets, it achieves high root-cause identification accuracy while yielding interpretable dependency structures.

Abstract

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