Learning Unknown Interdependencies for Decentralized Root Cause Analysis in Nonlinear Dynamical Systems

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.

Networks of interdependent industrial assets (clients) are tightly coupled through physical processes and control inputs, raising a key question: how would the output of one client change if another client were operated differently? This is difficult to answer because client-specific data are high-dimensional and private, making centralization of raw data infeasible. Each client also maintains proprietary local models that cannot be modified. We propose a federated framework for causal representation learning in state-space systems that captures interdependencies among clients under these constraints. Each client maps high-dimensional observations into low-dimensional latent states that disentangle intrinsic dynamics from control-driven influences. A central server estimates the global state-transition and control structure. This enables decentralized counterfactual reasoning where clients predict how outputs would change under alternative control inputs at others while only exchanging compact latent states. We prove convergence to a centralized oracle and provide privacy guarantees. Our experiments demonstrate scalability, and accurate cross-client counterfactual inference on synthetic and real-world industrial control system datasets.

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.