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Distributed Koopman Learning with Incomplete Measurements

Wenjian Hao, Lili Wang, Ayush Rai, Shaoshuai Mou

TL;DR

The paper addresses learning the dynamics of a nonlinear time-invariant system when multiple agents have only partial observations. It proposes a distributed deep Koopman learning framework (DDKL-PO) that combines Koopman lifting, deep observable functions, and consensus to estimate the global dynamics from locally observed data. Key contributions include a joint state-estimation and dynamics-learning algorithm, a two-part loss with diminishing-step updates, and neighbor communication of lifted observables to achieve performance close to centralized methods. Experiments on Lunar Lander demonstrate scalability to multi-agent networks with partial information while maintaining accuracy and reducing communication overhead.

Abstract

Koopman operator theory has emerged as a powerful tool for system identification, particularly for approximating nonlinear time-invariant systems (NTIS). This paper considers a network of agents with limited observation capabilities that collaboratively estimate the dynamics of an NTIS. A distributed deep Koopman learning algorithm is developed by integrating Koopman operator theory, deep neural networks, and consensus-based coordination. In the proposed framework, each agent approximates the system dynamics using its partial measurements and lifted states exchanged with its neighbors. This cooperative scheme enables accurate reconstruction of the global dynamics despite the absence of full-state information at individual agents. Simulation results on the Lunar Lander environment from OpenAI Gym demonstrate that the proposed method achieves performance comparable to the centralized deep Koopman learning with full-state access.

Distributed Koopman Learning with Incomplete Measurements

TL;DR

The paper addresses learning the dynamics of a nonlinear time-invariant system when multiple agents have only partial observations. It proposes a distributed deep Koopman learning framework (DDKL-PO) that combines Koopman lifting, deep observable functions, and consensus to estimate the global dynamics from locally observed data. Key contributions include a joint state-estimation and dynamics-learning algorithm, a two-part loss with diminishing-step updates, and neighbor communication of lifted observables to achieve performance close to centralized methods. Experiments on Lunar Lander demonstrate scalability to multi-agent networks with partial information while maintaining accuracy and reducing communication overhead.

Abstract

Koopman operator theory has emerged as a powerful tool for system identification, particularly for approximating nonlinear time-invariant systems (NTIS). This paper considers a network of agents with limited observation capabilities that collaboratively estimate the dynamics of an NTIS. A distributed deep Koopman learning algorithm is developed by integrating Koopman operator theory, deep neural networks, and consensus-based coordination. In the proposed framework, each agent approximates the system dynamics using its partial measurements and lifted states exchanged with its neighbors. This cooperative scheme enables accurate reconstruction of the global dynamics despite the absence of full-state information at individual agents. Simulation results on the Lunar Lander environment from OpenAI Gym demonstrate that the proposed method achieves performance comparable to the centralized deep Koopman learning with full-state access.
Paper Structure (7 sections, 19 equations, 4 figures, 1 algorithm)

This paper contains 7 sections, 19 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. The Problem
  3. Main Results
  4. Challenges and Key Ideas
  5. Algorithm
  6. Numerical Simulations
  7. Concluding Remarks

Figures (4)

  • Figure 1: Illustration of the partial observations.
  • Figure 2: Five-agent directed network with self-arcs, where self-arcs are omitted for simplicity.
  • Figure 3: Learning loss in \ref{['eq_loss_compact']} and estimation errors in \ref{['eq_Xmat_update']}.
  • Figure 4: Estimation errors on the training dataset (top) and testing dataset (bottom). The solid line denotes the mean error over five runs, and the shaded area indicates the standard deviation. Results are shown for DKO (green) and DDKL-PO (blue).

Theorems & Definitions (1)

  • Remark 1