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Distributed Koopman Operator Learning from Sequential Observations

Ali Azarbahram, Shenyu Liu, Gian Paolo Incremona

TL;DR

This work addresses learning nonlinear dynamics from distributed, sequential observations by formulating a Frobenius-norm Koopman regression that across agents enforces consensus on a common operator. It proposes a discrete-time PI-consensus algorithm that updates local Koopman matrices and an integral state, with convergence guarantees tied to the spectrum of a problem-dependent matrix and a step size bound. The results from a multi-UAV simulation show that the distributed solution recovers a global operator close to the centralized one and enables accurate short-horizon forecasting under sensing constraints. The framework preserves privacy and scalability while accommodating asynchronous sensing and limited inter-agent communication, suggesting practical applicability in resource-constrained multi-agent settings.

Abstract

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.

Distributed Koopman Operator Learning from Sequential Observations

TL;DR

This work addresses learning nonlinear dynamics from distributed, sequential observations by formulating a Frobenius-norm Koopman regression that across agents enforces consensus on a common operator. It proposes a discrete-time PI-consensus algorithm that updates local Koopman matrices and an integral state, with convergence guarantees tied to the spectrum of a problem-dependent matrix and a step size bound. The results from a multi-UAV simulation show that the distributed solution recovers a global operator close to the centralized one and enables accurate short-horizon forecasting under sensing constraints. The framework preserves privacy and scalability while accommodating asynchronous sensing and limited inter-agent communication, suggesting practical applicability in resource-constrained multi-agent settings.

Abstract

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.

Paper Structure

This paper contains 5 sections, 2 theorems, 33 equations, 8 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Statement
  3. Main Results
  4. Simulation results
  5. Conclusion

Key Result

Theorem 1

Suppose the communication graph $G$ is undirect and connected. For any gains $k_P,k_I>0$, there exists such that as long as the step size $\alpha<\alpha_{\max}$, the $K_i$ components of the algorithm will reach consensus and converge to an optimal solution of the problem eq:frobenius_minimization. Furthermore, the convergence is exponential with rate $\rho>\rho_{\max}$, where

Figures (8)

  • Figure 1: Conceptual illustration of sequential sensing and distributed Koopman learning.
  • Figure 2: Spectral comparison between $K_{\mathrm{ave}}$ and $K^*$.
  • Figure 3: Prediction error of distributed Koopman operators.
  • Figure 4: Convergence of the distributed Koopman learning objective
  • Figure 5: Consensus among distributed Koopman operators.
  • ...and 3 more figures

Theorems & Definitions (8)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Lemma 1
  • proof
  • proof : Proof of Theorem \ref{['thm:col']}