Distributed Koopman Operator Learning from Sequential Observations

This paper presents a distributed Koopman operator learning framework for modeling unknown nonlinear dynamics using sequential observations from multiple agents. Each agent estimates a local Koopman approximation based on lifted data and collaborates over a communication graph to reach exponential consensus on a consistent distributed approximation. The approach supports distributed computation under asynchronous and resource-constrained sensing. Its performance is demonstrated through simulation results, validating convergence and predictive accuracy under sensing-constrained scenarios and limited communication.

💡 Research Summary

The paper introduces a distributed learning framework for Koopman operators that operates under sequential, temporally fragmented observations collected by multiple agents. Traditional Koopman approximations rely on centralized processing of large data sets, which becomes infeasible in large‑scale, privacy‑sensitive, or bandwidth‑limited multi‑agent scenarios. The authors formulate the problem as a Frobenius‑norm least‑squares minimization of the Koopman matrix (K) subject to a consensus constraint across agents. Each agent (i) possesses only its local data block ((X_i, Y_i)) and seeks a local estimate (K_i) that minimizes (|Y_i - K_i X_i|_F^2) while ensuring that all (K_i) agree. By stacking the local variables and using the graph Laplacian (L), the consensus condition is expressed compactly as (K L = 0), leading to a global optimization problem with linear equality constraints.

To solve this problem in a fully distributed manner, the authors propose a discrete‑time algorithm based on a proportional‑integral (PI) consensus law. The update for each agent consists of three components: (1) a gradient‑descent step on its local least‑squares cost, (2) a proportional diffusion term that drives the local estimate toward the average of its neighbors, and (3) an integral term that accumulates the diffusion error over time, thereby guaranteeing asymptotic agreement. The update equations are: \