Federated Causal Representation Learning in State-Space Systems for Decentralized Counterfactual Reasoning

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.

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

Networks of modern industrial systems are increasingly monitored by distributed sensors, where each system comprises multiple subsystems generating high dimensional time series data. These subsystems are often interdependent, making it important to understand how temporal patterns at one subsystem relate to others. This is challenging in decentralized settings where raw measurements cannot be shared and client observations are heterogeneous. In practical deployments each subsystem (client) operates a fixed proprietary model that cannot be modified or retrained, limiting existing approaches. Nonlinear dynamics further make cross client temporal interdependencies difficult to interpret because they are embedded in nonlinear state transition functions. We present a federated framework for learning temporal interdependencies across clients under these constraints. Each client maps high dimensional local observations to low dimensional latent states using a nonlinear state space model. A central server learns a graph structured neural state transition model over the communicated latent states using a Graph Attention Network. For interpretability we relate the Jacobian of the learned server side transition model to attention coefficients, providing the first interpretable characterization of cross client temporal interdependencies in decentralized nonlinear systems. We establish theoretical convergence guarantees to a centralized oracle and validate the framework through synthetic experiments demonstrating convergence, interpretability, scalability and privacy. Additional real world experiments show performance comparable to decentralized baselines.

Uncertainty in Federated Granger Causality: From Origins to Systemic Consequences

Granger Causality (GC) provides a rigorous framework for learning causal structures from time-series data. Recent federated variants of GC have targeted distributed infrastructure applications (e.g., smart grids) with distributed clients that generate high-dimensional data bound by data-sovereignty constraints. However, Federated GC algorithms only yield deterministic point estimates of causality and neglect uncertainty. This paper establishes the first methodology for rigorously quantifying uncertainty and its propagation within federated GC frameworks. We systematically classify sources of uncertainty, explicitly differentiating aleatoric (data noise) from epistemic (model variability) effects. We derive closed-form recursions that model the evolution of uncertainty through client-server interactions and identify four novel cross-covariance components that couple data uncertainties with model parameter uncertainties across the federated architecture. We also define rigorous convergence conditions for these uncertainty recursions and obtain explicit steady-state variances for both server and client model parameters. Our convergence analysis demonstrates that steady-state variances depend exclusively on client data statistics, thus eliminating dependence on initial epistemic priors and enhancing robustness. Empirical evaluations on synthetic benchmarks and real-world industrial datasets demonstrate that explicitly characterizing uncertainty significantly improves the reliability and interpretability of federated causal inference.

Federated Granger Causality Learning for Interdependent Clients with State Space Representation

Advanced sensors and IoT devices have improved the monitoring and control of complex industrial enterprises. They have also created an interdependent fabric of geographically distributed process operations (clients) across these enterprises. Granger causality is an effective approach to detect and quantify interdependencies by examining how one client's state affects others over time. Understanding these interdependencies captures how localized events, such as faults and disruptions, can propagate throughout the system, possibly causing widespread operational impacts. However, the large volume and complexity of industrial data pose challenges in modeling these interdependencies. This paper develops a federated approach to learning Granger causality. We utilize a linear state space system framework that leverages low-dimensional state estimates to analyze interdependencies. This addresses bandwidth limitations and the computational burden commonly associated with centralized data processing. We propose augmenting the client models with the Granger causality information learned by the server through a Machine Learning (ML) function. We examine the co-dependence between the augmented client and server models and reformulate the framework as a standalone ML algorithm providing conditions for its sublinear and linear convergence rates. We also study the convergence of the framework to a centralized oracle model. Moreover, we include a differential privacy analysis to ensure data security while preserving causal insights. Using synthetic data, we conduct comprehensive experiments to demonstrate the robustness of our approach to perturbations in causality, the scalability to the size of communication, number of clients, and the dimensions of raw data. We also evaluate the performance on two real-world industrial control system datasets by reporting the volume of data saved by decentralization.