Distributed Koopman Operator Learning from Sequential Observations

Ali Azarbahram; Gian Paolo Incremona; Shenyu Liu

arxiv: 2509.20071 · v2 · submitted 2025-09-24 · 📡 eess.SY · cs.SY

Distributed Koopman Operator Learning from Sequential Observations

Ali Azarbahram , Shenyu Liu , Gian Paolo Incremona This is my paper

Pith reviewed 2026-05-18 14:45 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords Koopman operatordistributed learningmulti-agent systemsconsensus algorithmsnonlinear dynamicssequential observations

0 comments

The pith

Agents with partial views of nonlinear dynamics reach consensus on a shared Koopman model from local sequential data.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a distributed framework in which each agent builds its own local Koopman approximation from the sequence of observations it collects. These local models are then aligned through a consensus process that runs over the agents' communication graph, producing exponential convergence to one consistent global approximation. The method is designed for cases where sensing is asynchronous and communication is limited, so that no central node needs to gather every measurement. Simulations illustrate that the resulting distributed model converges and maintains predictive accuracy under these constraints.

Core claim

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 of the unknown nonlinear dynamics.

What carries the argument

Exponential consensus protocol over the communication graph that synchronizes the local Koopman operator matrices each agent computes from its lifted sequential observations.

If this is right

The approach supports distributed computation under asynchronous and resource-constrained sensing.
It achieves convergence and predictive accuracy in simulations with limited communication.
A consistent global model of the nonlinear dynamics is obtained without central aggregation of raw observations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same consensus structure could be tested on time-varying communication graphs that appear or disappear during operation.
It suggests a route for teams of robots to maintain a shared model of an environment even when each robot sees only a narrow slice of the space.
Adding explicit handling of measurement noise would be a direct next step to check robustness.

Load-bearing premise

The local approximations each agent computes from its own partial observations are similar enough that their network consensus recovers an accurate description of the full system dynamics.

What would settle it

A test in which agents receive completely disjoint sequences of observations and the converged consensus model shows substantially higher prediction error on held-out trajectories than a single model built from all data together.

Figures

Figures reproduced from arXiv: 2509.20071 by Ali Azarbahram, Gian Paolo Incremona, Shenyu Liu.

**Figure 1.** Figure 1: Spectral comparison between Kave and K∗ . G encoded by Laplacian L, reflecting decentralized conditions. Each agent runs 1000 iterations of the distributed update, using kP = 150, kI = 75, and α = 0.5αmax with αmax = 0.03 from (14). This ensures exponential convergence with ρmax = 0.92 as per (15). We analyze the spectral similarity between centralized and distributed Koopman operators. Let K∗ be the cent… view at source ↗

**Figure 2.** Figure 2: Elementwise absolute difference between Kave and K∗ . structures matching spatially-local observables on the 20×20 grid. Minor deviations (around 10−4 ) likely stem from observability gaps or communication limits, demonstrating the method’s resilience and offering insight for future refinements. To assess accuracy, we compute 1/pPp i=1 kY − KiXkF , measuring how well each distributed Koopman operator Ki … view at source ↗

**Figure 3.** Figure 3: Prediction error of distributed Koopman operator [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

**Figure 4.** Figure 4: Heatmap of distributed Koopman prediction error o [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper extends Koopman learning to multi-agent networks by having each agent compute a local approximation then reach consensus over a graph, but the step from partial local data to accurate global model rests mostly on simulations without explicit consistency bounds.

read the letter

The core contribution is a distributed framework where agents each lift their sequential observations to build local Koopman approximations and then apply exponential consensus over a communication graph to agree on a shared model. This handles asynchronous and resource-limited sensing in networked systems, which is a practical step beyond centralized DMD-style methods. The simulations reportedly show convergence and decent predictive accuracy under constrained conditions, giving some evidence that the approach can work in multi-agent settings. Credit is due for framing the problem clearly and demonstrating the consensus step in simulation. The main soft spot is the load-bearing assumption that independently computed local operators remain close enough for consensus to recover a faithful global approximation. When observations are partial or non-overlapping, the data matrices at each node can differ in both trajectory coverage and the effective span of the lifted space, yet the paper does not appear to supply bounds on the resulting operator discrepancy in terms of overlap, graph connectivity, or lifting uniformity. Without that, the transition from local estimates to global fidelity is not fully secured beyond the reported runs. The math relies on standard consensus and Koopman lifting, which is fine, but tighter error analysis would strengthen the claims. This work is aimed at researchers in distributed control and data-driven modeling of networked nonlinear systems. A reader looking for a concrete distributed implementation with simulation backing would find it useful. It deserves peer review because the idea addresses a real scalability need and the simulations provide a starting point, even if the theoretical gaps around consistency need referee scrutiny.

Referee Report

2 major / 2 minor

Summary. The paper proposes a distributed framework for Koopman operator learning from sequential observations collected by multiple agents. Each agent computes a local Koopman approximation via lifted data and DMD-style regression; these local operators are then exchanged over a communication graph using a consensus protocol that achieves exponential convergence to a shared global approximation. The method targets asynchronous, sensing-constrained, and resource-limited multi-agent settings, with performance illustrated by simulation results on convergence and predictive accuracy.

Significance. If the local-to-global consistency holds, the approach would enable scalable, privacy-preserving identification of nonlinear dynamics without central data aggregation, which is relevant for sensor networks, robotic swarms, and distributed control. The combination of Koopman lifting with standard consensus is a natural extension, and the emphasis on asynchronous updates addresses practical constraints. However, the current validation rests entirely on simulations mentioned in the abstract, so the result's impact depends on whether theoretical guarantees or reproducible code can be added.

major comments (2)

[Local Estimation and Distributed Consensus (likely §3–4)] The central claim that consensus over the graph recovers an accurate global model rests on the unstated assumption that independently computed local Koopman operators (from partial, possibly non-overlapping trajectories) remain sufficiently close. No explicit bounds are given on the operator discrepancy induced by differing observed trajectories or spans of the lifted space; this discrepancy is load-bearing because the consensus limit equals the centralized operator only if the local estimates satisfy a uniform closeness condition that is not derived.
[Convergence Analysis] The exponential-consensus result appears to invoke standard linear consensus theorems without adjusting for the fact that each agent's local operator is recomputed from streaming, partial data and may therefore be time-varying or inconsistent across nodes. A concrete condition relating graph connectivity, observation overlap, and lifting-function uniformity to the consensus error is required to support the claim of global predictive accuracy.

minor comments (2)

[Numerical Results] Expand the simulation section with explicit quantitative metrics (e.g., operator-norm error, prediction horizon RMSE) and details on the number of agents, graph topology, and data-exclusion rules so that the reported convergence can be reproduced.
[Preliminaries] Clarify notation for the lifted observables and the precise regression problem solved at each node; the transition from local data matrices to the operator used in consensus should be written with explicit matrix dimensions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the insightful comments, which highlight important aspects of the theoretical foundations. We address each major comment below and will revise the manuscript accordingly to strengthen the analysis.

read point-by-point responses

Referee: [Local Estimation and Distributed Consensus (likely §3–4)] The central claim that consensus over the graph recovers an accurate global model rests on the unstated assumption that independently computed local Koopman operators (from partial, possibly non-overlapping trajectories) remain sufficiently close. No explicit bounds are given on the operator discrepancy induced by differing observed trajectories or spans of the lifted space; this discrepancy is load-bearing because the consensus limit equals the centralized operator only if the local estimates satisfy a uniform closeness condition that is not derived.

Authors: We agree that the manuscript does not derive explicit bounds on the discrepancy between local Koopman operators arising from partial or non-overlapping trajectories. In the revised version, we will add a new lemma establishing such bounds under assumptions of sufficient trajectory overlap and uniform lifting functions across agents. This will make precise the condition under which the consensus limit coincides with the centralized operator. revision: yes
Referee: [Convergence Analysis] The exponential-consensus result appears to invoke standard linear consensus theorems without adjusting for the fact that each agent's local operator is recomputed from streaming, partial data and may therefore be time-varying or inconsistent across nodes. A concrete condition relating graph connectivity, observation overlap, and lifting-function uniformity to the consensus error is required to support the claim of global predictive accuracy.

Authors: The current convergence argument applies standard linear consensus results directly to the sequence of local estimates. We acknowledge that this does not explicitly account for the time-varying nature induced by streaming data. In the revision, we will extend the analysis to provide a concrete sufficient condition linking graph connectivity, minimum observation overlap, and lifting-function uniformity to an explicit bound on the consensus error, thereby supporting the global predictive accuracy claim. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard methods

full rationale

The paper presents a distributed framework in which agents independently compute local Koopman approximations via lifting and regression on their partial sequential data, then apply standard consensus protocols over a communication graph to converge on a shared operator. No step in the described chain reduces a claimed prediction or global model to a quantity defined by the same fitted parameters or by self-referential equations; the local estimates and consensus update are treated as independent algorithmic components whose consistency is an external assumption rather than a derived identity. The abstract and framework description invoke established Koopman lifting and exponential consensus results without renaming known patterns or smuggling ansatzes through self-citations that would force the outcome by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard assumptions from graph theory and Koopman operator theory; no new entities are postulated and free parameters are not enumerated in the abstract.

axioms (1)

domain assumption The communication graph is connected and permits exponential consensus.
Invoked to guarantee that local estimates converge to a single consistent approximation.

pith-pipeline@v0.9.0 · 5591 in / 1161 out tokens · 61654 ms · 2026-05-18T14:45:38.455152+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in ":" * " " * FUNCTION f...

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" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in ":" * " " * FUNCTION f...

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" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in ":" * " " * FUNCTION f...

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author Cao, Y. , author Yu, W. , author Ren, W. , author Chen, G. , year 2012 . title An overview of recent progress in the study of distributed multi-agent coordination . journal IEEE Transactions on Industrial informatics volume 9 , pages 427--438

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author Hao, W. , author Lu, Z. , author Upadhyay, D. , author Mou, S. , year 2024 a. title A distributed deep K oopman learning algorithm for control . journal arXiv preprint arXiv:2412.07212

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, author Nandanoori, S.P

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, author Pal, S

author Nandanoori, S.P. , author Pal, S. , author Sinha, S. , author Kundu, S. , author Agarwal, K. , author Choudhury, S. , year 2021 . title Data-driven distributed learning of multi-agent systems: A K oopman operator approach , in: booktitle 2021 60th IEEE Conference on Decision and Control (CDC) , organization IEEE . pp. pages 5059--5066

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, author Lee, M.H

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, author Slotine, J.J

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, author Mou, S

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, author Zhou, J

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work page doi:10.1109/tac.2019.2912533 2019

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author Yang, T. , author George, J. , author Qin, J. , author Yi, X. , author Wu, J. , year 2020 . title Distributed least squares solver for network linear equations . journal Automatica volume 113 , pages 108798 . :https://doi.org/10.1016/j.automatica.2019.108798

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