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.

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