Koopman Identification of Nonlinear Systems via Reservoir Liftings

Chen Yang; Lu Shi; Weibin Gu

arxiv: 2605.04917 · v1 · submitted 2026-05-06 · 💻 cs.LG · cs.RO

Koopman Identification of Nonlinear Systems via Reservoir Liftings

Weibin Gu , Chen Yang , Lu Shi This is my paper

Pith reviewed 2026-05-08 18:18 UTC · model grok-4.3

classification 💻 cs.LG cs.RO

keywords Koopman operatorreservoir computingnonlinear system identificationdynamical systemsecho state propertylinear embedding

0 comments

The pith

Reservoir computing supplies a memory-tuned dictionary that lifts nonlinear dynamics into a stable linear Koopman form.

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

The paper introduces a method that treats a reservoir computer as a finite collection of time-dependent features whose memory length is set by the reservoir's spectral radius. This construction turns the problem of finding a linear representation of a nonlinear system into a well-conditioned regression whose conditioning is guaranteed by the echo state property. A simple correlation rule then tunes the memory length to match the dominant time scales present in the data. The resulting linear model reconstructs the original trajectories more faithfully than standard dictionary methods while preserving long-term stability on the tested examples.

Core claim

The RC-Koopman framework interprets the reservoir as a stateful, finite-dimensional Koopman dictionary whose temporal depth is set by its spectral radius. The echo state property ensures the lifted features produce a well-posed linear operator. A correlation-based procedure aligns that depth with the system's dominant time scales, which in turn determines which Koopman eigenfunctions remain observable from the lifted states.

What carries the argument

The reservoir viewed as a stateful Koopman dictionary whose memory depth is controlled by spectral radius.

If this is right

Finite reservoir memory selects a subset of Koopman eigenfunctions that can be recovered from the lifted observations.
The lifted linear model achieves higher trajectory reconstruction accuracy while maintaining dynamical stability on synthetic benchmarks.
The method avoids the dictionary-selection and ill-conditioning issues that affect extended dynamic mode decomposition and Hankel lifting.
Spectral-radius tuning directly trades off observable memory depth against numerical stability of the recovered operator.

Where Pith is reading between the lines

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

The same memory-alignment rule could be applied to streaming data to adapt the reservoir on the fly without retraining.
Extending the reservoir to multiple parallel spectral radii might recover a wider set of eigenfunctions for systems with widely separated time scales.
The framework suggests a route to embedding the lifted linear model inside a controller whose stability margins are directly inherited from the reservoir property.

Load-bearing premise

The echo state property of the reservoir guarantees that the lifted feature matrix remains well-conditioned for linear regression.

What would settle it

Replace the correlation-based spectral-radius choice with a deliberately mismatched value and measure whether the linear model's long-term prediction error rises while its stability margin falls on the same benchmark trajectories.

Figures

Figures reproduced from arXiv: 2605.04917 by Chen Yang, Lu Shi, Weibin Gu.

**Figure 1.** Figure 1: Schematic of the proposed RC–Koopman framework. System measurements are mapped to a high-dimensional reservoir that serves as a stateful dictionary, where the resulting reservoir state is augmented by the system state or output to form the lifted state ψk . The Koopman matrices A and B are then identified via least-squares regression using time-shifted snapshot matrices (Ψ, Ψ′ ) and control inputs (U). sub… view at source ↗

**Figure 2.** Figure 2: Comparison of ground truth and reconstructed dynamics. (a) The Duffing oscillator and (b) a differential-drive robot model. TABLE I NRMSE COMPARISON FOR RECONSTRUCTION. Method NRMSE Duffing Oscillator Differential-Drive Robot RC–Koopman 5.38 × 10−3 2.02 × 10−1 EDMD 3.17 × 10−2 2.02 × 10−1 HAVOK 7.39 × 10−4 2.03 × 10−1 the Duffing oscillator, reservoir lifting satisfies the persistent excitation condition … view at source ↗

**Figure 3.** Figure 3: Eigenvalue spectra of the learned Koopman operators. (a) The Duffing oscillator and (b) a differential-drive robot. Dots (•) indicate eigenvalues within or on the unit circle, while crosses (×) denote eigenvalues outside the unit circle, representing numerically unstable modes. For visual clarity, eigenvalues with |λ| > 3 are not shown view at source ↗

**Figure 4.** Figure 4: Koopman eigenvalue lifetimes Ti versus reservoir memory horizon τϵ across varying spectral radii ρ. Each dot represents an identified eigenvalue, color-coded by ρ. The diagonal line Ti = τϵ indicates the boundary of identifiable dynamics: eigenvalues below the line evolve on time scales shorter than the reservoir memory and are therefore observable in the lifted representation. The spectral radius selected… view at source ↗

read the original abstract

Learning tractable linear representations of nonlinear dynamical systems via Koopman operator theory is often hindered by dictionary selection, temporal memory encoding, and numerical ill-conditioning. Inspired by Reservoir Computing (RC) paradigm, this paper introduces the RC-Koopman framework, which interprets reservoir as a stateful, finite-dimensional Koopman dictionary whose temporal depth is explicitly controlled by its spectral radius. We show that the Echo State Property (ESP) guarantees well-posedness and favorable numerical conditioning of the lifted Koopman approximation. A correlation-based spectral radius selection algorithm aligns reservoir memory with dominant system timescales. Analysis reveals how the finite memory of the reservoir determines which Koopman eigenfunctions remain observable from the lifted features. Evaluation on synthetic benchmarks demonstrates that RC-Koopman achieves a favorable balance between reconstruction accuracy of the underlying nonlinear dynamics and dynamical stability, compared to Extended Dynamic Mode Decomposition (EDMD) and Hankel-based lifting approaches. Code available at: https://github.com/NEAR-the-future/RC-Koopman.git

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

RC-Koopman uses reservoirs for memory-tuned Koopman lifts but needs stronger checks on its benchmark claims.

read the letter

Here's the quick read on this one. The paper's core move is to use a reservoir computing setup as the lifting function for Koopman identification, where the spectral radius directly sets how far back the memory goes. They add a correlation-based rule to pick that radius so it lines up with the main timescales in the data, and they argue that the echo state property keeps the whole thing numerically stable. What feels new is the analysis showing how the reservoir's finite memory affects observability of the Koopman eigenfunctions. That part connects the RC side to the Koopman side in a way that goes a bit beyond just calling the reservoir a dictionary. The practical algorithm for radius selection is also a useful detail if it works reliably. On the positive side, the synthetic benchmarks suggest RC-Koopman can recover the dynamics more accurately than basic EDMD while keeping better stability than Hankel approaches. The public code is a plus for anyone who wants to try it out. The weaker parts are in how much we can trust those benchmark results. Without error bars, details on data splits, or checks on whether the spectral radius choice was fixed before seeing the outcomes, it's hard to know if the favorable balance holds up or if it's helped by the selection step. The concern about the correlation method potentially biasing toward the observed data seems real and should be addressed with more controls or cross-validation. This is the kind of paper that fits in a reading group for people doing data-driven dynamics or nonlinear control. Someone already familiar with both RC and EDMD would see the value quickly and might adapt the radius trick. I think it deserves peer review. The framing is solid enough and the claims are specific enough that referees can check them directly.

Referee Report

3 major / 2 minor

Summary. The paper proposes the RC-Koopman framework that treats a reservoir computing reservoir as a stateful, finite-dimensional Koopman dictionary whose temporal memory is controlled by the spectral radius. It invokes the Echo State Property (ESP) to guarantee well-posedness and numerical conditioning of the lifted approximation, introduces a correlation-based algorithm to select the spectral radius so that reservoir memory aligns with dominant system timescales, analyzes how finite reservoir memory determines observability of Koopman eigenfunctions, and reports that the resulting method achieves a favorable accuracy-stability trade-off on synthetic benchmarks relative to EDMD and Hankel-based lifting.

Significance. If the central claims hold, the work supplies a practical, theoretically grounded route to constructing Koopman dictionaries that incorporate controllable memory and stability properties, bridging reservoir computing with operator-theoretic system identification. The explicit analysis of eigenfunction observability under finite memory is a useful contribution. The empirical demonstration of balanced performance is potentially valuable for applications, but its impact is limited by the absence of statistical validation and by the data-dependent nature of the spectral-radius procedure.

major comments (3)

[§3.3] §3.3 (Correlation-based spectral radius selection): The algorithm selects the spectral radius by maximizing sample correlations between reservoir states and observed trajectories. This procedure can implicitly tune the reservoir memory to the particular finite-length, possibly noisy realization rather than to intrinsic timescales, creating a risk that the reported accuracy-stability balance is partly an artifact of post-hoc fitting; a concrete test would be to fix the radius by an independent criterion (e.g., theoretical ESP bound) and re-evaluate the benchmarks.
[§5] §5 (Experimental evaluation): The synthetic-benchmark results are presented without error bars, without explicit data-exclusion rules, and without sensitivity sweeps over the chosen spectral radii. Consequently it is impossible to judge whether the claimed superiority over EDMD and Hankel methods is statistically robust or sensitive to the post-hoc radius selection.
[§2.2 and §4] §2.2 and §4 (ESP guarantees): The manuscript asserts that the Echo State Property ensures well-posedness and favorable conditioning of the lifted Koopman operator, yet no explicit verification (e.g., numerical check of the ESP condition or condition-number bounds for the chosen reservoirs) is supplied for the experimental instances; without this, the stability half of the central claim rests on an unverified assumption.

minor comments (2)

Notation for the lifted feature map and the correlation objective could be made more explicit (e.g., distinguish reservoir state dimension from the number of retained Koopman modes).
A few references to recent reservoir-computing literature on spectral-radius tuning and ESP verification appear to be missing.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major point below, providing clarifications and indicating the revisions we will incorporate to strengthen the presentation and empirical support.

read point-by-point responses

Referee: [§3.3] §3.3 (Correlation-based spectral radius selection): The algorithm selects the spectral radius by maximizing sample correlations between reservoir states and observed trajectories. This procedure can implicitly tune the reservoir memory to the particular finite-length, possibly noisy realization rather than to intrinsic timescales, creating a risk that the reported accuracy-stability balance is partly an artifact of post-hoc fitting; a concrete test would be to fix the radius by an independent criterion (e.g., theoretical ESP bound) and re-evaluate the benchmarks.

Authors: The correlation-based procedure is intended to align reservoir memory with the dominant timescales of the observed dynamics, which is a core motivation of the RC-Koopman framework rather than arbitrary fitting. Nevertheless, we agree that an independent-criterion comparison would improve robustness. In the revised manuscript we will add experiments that fix the spectral radius via the theoretical ESP bound (e.g., ρ = 0.9) and re-evaluate the same synthetic benchmarks, allowing direct comparison with the adaptive selection. revision: yes
Referee: [§5] §5 (Experimental evaluation): The synthetic-benchmark results are presented without error bars, without explicit data-exclusion rules, and without sensitivity sweeps over the chosen spectral radii. Consequently it is impossible to judge whether the claimed superiority over EDMD and Hankel methods is statistically robust or sensitive to the post-hoc radius selection.

Authors: We accept that the current experimental section lacks sufficient statistical detail. The revision will include error bars computed over multiple independent reservoir initializations and data realizations, explicit statements of data-splitting and exclusion criteria, and sensitivity sweeps over spectral-radius values around the selected operating points. These additions will demonstrate that the reported accuracy-stability trade-off is not unduly sensitive to the radius choice. revision: yes
Referee: [§2.2 and §4] §2.2 and §4 (ESP guarantees): The manuscript asserts that the Echo State Property ensures well-posedness and favorable conditioning of the lifted Koopman operator, yet no explicit verification (e.g., numerical check of the ESP condition or condition-number bounds for the chosen reservoirs) is supplied for the experimental instances; without this, the stability half of the central claim rests on an unverified assumption.

Authors: Sections 2.2 and 4 prove that the ESP implies well-posedness and favorable conditioning of the lifted operator. In the experiments the spectral radius is deliberately kept below unity to satisfy the ESP. To make this explicit, the revised experimental section will report the numerical spectral-radius values used together with condition-number bounds on the lifted feature matrices for each benchmark, thereby verifying the conditioning claim for the chosen reservoirs. revision: yes

Circularity Check

0 steps flagged

No circularity: method choices and external properties support benchmarked claims

full rationale

The paper defines RC-Koopman by interpreting the reservoir as a finite-memory Koopman dictionary whose depth is set by spectral radius, selects that radius via an explicit correlation algorithm to align with observed timescales, and invokes the Echo State Property as an external guarantee for well-posedness and conditioning. The analysis of observable eigenfunctions follows directly from the finite-memory construction without reducing to a fitted input renamed as a prediction. The central claim of favorable accuracy-stability balance is demonstrated via direct comparison on synthetic benchmarks against EDMD and Hankel baselines, which are independent external references. No self-citations, ansatzes smuggled via prior work, or self-definitional loops appear in the derivation; all load-bearing elements remain self-contained against the stated benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Framework rests on reservoir computing assumptions and the Echo State Property as a domain guarantee; spectral radius selection is algorithmic but data-dependent.

free parameters (1)

spectral radius
Selected via correlation-based algorithm to align with dominant timescales; acts as a tunable hyperparameter controlling memory depth.

axioms (1)

domain assumption Echo State Property guarantees well-posedness and favorable numerical conditioning
Invoked directly to ensure the lifted Koopman approximation is stable and well-conditioned.

pith-pipeline@v0.9.0 · 5468 in / 1206 out tokens · 41480 ms · 2026-05-08T18:18:41.320188+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 1 canonical work pages · 1 internal anchor

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N. Boddupalli, A. Hasnain, S. P. Nandanoori, and E. Yeung, “Koopman operators for generalized persistence of excitation conditions for nonlinear systems,” in2019 IEEE 58th Conference on Decision and Control (CDC), pp. 8106–8111, 2019

2019

[1] [1]

Hamiltonian systems and transformation in Hilbert space,

B. O. Koopman, “Hamiltonian systems and transformation in Hilbert space,”Proceedings of the National Academy of Sciences, vol. 17, no. 5, pp. 315–318, 1931

1931

[2] [2]

Modern Koopman theory for dynamical systems,

S. L. Brunton, M. Budi ˇsi´c, E. Kaiser, and J. N. Kutz, “Modern Koopman theory for dynamical systems,”SIAM Review, 2022

2022

[3] [3]

Koopman operators in robot learn- ing,

L. Shi, M. Haseli, G. Mamakoukas, D. Bruder, I. Abraham, T. Mur- phey, J. Cort ´es, and K. Karydis, “Koopman operators in robot learn- ing,”IEEE Transactions on Robotics, 2026

2026

[4] [4]

A data– driven approximation of the Koopman operator: Extending dynamic mode decomposition,

M. O. Williams, I. G. Kevrekidis, and C. W. Rowley, “A data– driven approximation of the Koopman operator: Extending dynamic mode decomposition,”Journal of Nonlinear Science, vol. 25, no. 6, pp. 1307–1346, 2015

2015

[5] [5]

Chaos as an intermittently forced linear system,

S. L. Brunton, B. W. Brunton, J. L. Proctor, E. Kaiser, and J. N. Kutz, “Chaos as an intermittently forced linear system,”Nature Communications, vol. 8, no. 1, p. 19, 2017

2017

[6] [6]

Time-delay observables for Koopman: Theory and applications,

M. Kamb, E. Kaiser, S. L. Brunton, and J. N. Kutz, “Time-delay observables for Koopman: Theory and applications,”SIAM Journal on Applied Dynamical Systems, vol. 19, no. 2, pp. 886–917, 2020

2020

[7] [7]

Structured time-delay models for dynamical systems with connections to Frenet–Serret frame,

S. M. Hirsh, S. M. Ichinaga, S. L. Brunton, J. Nathan Kutz, and B. W. Brunton, “Structured time-delay models for dynamical systems with connections to Frenet–Serret frame,”Proceedings of the Royal Society A, vol. 477, no. 2254, p. 20210097, 2021

2021

[8] [8]

Reservoir computing approaches to recurrent neural network training,

M. Luko ˇseviˇcius and H. Jaeger, “Reservoir computing approaches to recurrent neural network training,”Computer Science Review, vol. 3, no. 3, pp. 127–149, 2009

2009

[9] [9]

Generic and isometric embeddings in reservoir com- puters,

A. G. Hart, “Generic and isometric embeddings in reservoir com- puters,”Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 35, no. 11, 2025

2025

[10] [10]

On Dominant Manifolds in Reservoir Computing Networks

N. Kaplan, A. Padoan, and A. Bizyaeva, “On dominant manifolds in reservoir computing networks,”arXiv preprint arXiv:2604.05967, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026

[11] [11]

The “echo state

H. Jaeger, “The “echo state” approach to analysing and training recurrent neural networks-with an erratum note,”Bonn, Germany: German national research center for information technology gmd technical report, vol. 148, no. 34, p. 13, 2001

2001

[12] [12]

On explaining the surprising success of reservoir computing forecaster of chaos? The universal machine learning dynamical system with contrast to V AR and DMD,

E. Bollt, “On explaining the surprising success of reservoir computing forecaster of chaos? The universal machine learning dynamical system with contrast to V AR and DMD,”Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 31, no. 1, 2021

2021

[13] [13]

Next generation reservoir computing,

D. J. Gauthier, E. Bollt, A. Griffith, and W. A. Barbosa, “Next generation reservoir computing,”Nature Communications, vol. 12, no. 1, p. 5564, 2021

2021

[14] [14]

Two methods to approximate the Koopman operator with a reservoir computer,

M. Gulina and A. Mauroy, “Two methods to approximate the Koopman operator with a reservoir computer,”Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 31, no. 2, 2021

2021

[15] [15]

Model reduction of dynamical systems with a novel data-driven approach: The RC- HA VOK algorithm,

G. Yılmaz Bing ¨ol, O. Soysal, and E. G ¨unay, “Model reduction of dynamical systems with a novel data-driven approach: The RC- HA VOK algorithm,”Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 34, no. 8, 2024

2024

[16] [16]

Dynamic mode decomposition with control,

J. L. Proctor, S. L. Brunton, and J. N. Kutz, “Dynamic mode decomposition with control,”SIAM Journal on Applied Dynamical Systems, vol. 15, no. 1, pp. 142–161, 2016

2016

[17] [17]

Koopman operators for generalized persistence of excitation conditions for nonlinear systems,

N. Boddupalli, A. Hasnain, S. P. Nandanoori, and E. Yeung, “Koopman operators for generalized persistence of excitation conditions for nonlinear systems,” in2019 IEEE 58th Conference on Decision and Control (CDC), pp. 8106–8111, 2019

2019