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arxiv: 2507.09646 · v4 · submitted 2025-07-13 · 📡 eess.SY · cs.SY

Learning Koopman Models From Data Under General Noise Conditions

Lucian Cristian Iacob , M\'at\'e Sz\'ecsi , Gerben Izaak Beintema , Maarten Schoukens , Roland T\'oth This is my paper

Pith reviewed 2026-05-19 05:00 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords koopman modelssystem identificationnonlinear dynamicsdeep learningmultiple shootinginnovation noisestatistical consistencyinput-output data
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The pith

A multiple-shooting formulation with deep encoders yields statistically consistent Koopman models from input-output data under general noise.

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

The paper develops an identification technique for Koopman representations of nonlinear dynamical systems that have inputs and are subject to general noise. It relies on deep state-space encoders to reconstruct the lifted state from input-output sequences and minimizes a prediction error loss defined through multiple shooting. An innovation noise term is added to the model to account for disturbances in both process and measurements. A sympathetic reader would care because this promises models that become arbitrarily accurate with more data while remaining practical to compute and good at forecasting far into the future.

Core claim

The authors establish that their approach, which combines deep state-space encoders based on state reconstructability with a multiple-shooting squared loss that includes an innovation noise term, produces Koopman models whose parameters can be estimated consistently from input-output data alone, with the error vanishing as the data volume grows, while the parallelizable formulation supports efficient optimization and strong long-horizon predictions.

What carries the argument

Deep state-space encoders that exploit the state reconstructability property to recover the Koopman lifted state, paired with a multiple-shooting formulation of the prediction error loss and an innovation noise model.

If this is right

  • The estimator is statistically consistent, so estimation error goes to zero with infinite data.
  • Multiple-shooting enables parallel computation of the loss over data segments for faster batch training.
  • Obtained models achieve excellent long-term prediction on nonlinear benchmarks and quadcopter flight data.

Where Pith is reading between the lines

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

  • Such models could support predictive control without requiring direct state sensors.
  • Similar techniques might extend to other lifted representations beyond Koopman.
  • Validation on systems with structured noise could show where the general innovation term suffices or needs refinement.

Load-bearing premise

The nonlinear system must allow state reconstruction from inputs and outputs through the deep encoder, and the single innovation noise term must sufficiently describe all process and measurement disturbances.

What would settle it

If increasing the amount of training data from the quadcopter or benchmark systems does not reduce the long-term prediction error or the parameter estimation discrepancy, the statistical consistency would be refuted.

Figures

Figures reproduced from arXiv: 2507.09646 by Gerben Izaak Beintema, Lucian Cristian Iacob, Maarten Schoukens, M\'at\'e Sz\'ecsi, Roland T\'oth.

Figure 1
Figure 1. Figure 1: Compliant surface corresponding to Ψ (in blue), compliant trajectories of the lifted [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Network architecture. The lifted state at moment [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Wiener-Hammerstein system Theorem 4.2. Under the conditions of Theorem 4.1 and Condition 3, lim N→∞ ˆ (4.6) ξN ∈ Ξo with probability 1, where ˆξN = arg min vec(θ,η)∈Ξ V enc DN (4.7) (θ, η). Proof. The proof is a direct application of Lemma 4.1 in [21] because the loss function (3.18a) fulfills Condition (4.4) in [21]. 5. Experiments and results. Next we test the proposed Koopman model identification approa… view at source ↗
Figure 4
Figure 4. Figure 4: NRMS of the simulation responses of the process part of the Koopman [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: NRMS of the one-step-ahead prediction by the Koopman models [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Train, validation and test (multisine, sine and sinesweep) datasets used for the [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Overview of NRMS errors of identified Koopman models with different complexities [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Simulated output responses of the estimated model ( [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Crazyflie 2.1 during flight [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Simulation results of the estimated Koopman model and the estimated nonlinear [PITH_FULL_IMAGE:figures/full_fig_p025_10.png] view at source ↗
read the original abstract

This paper presents a novel identification approach of Koopman models of nonlinear systems with inputs under rather general noise conditions. The method uses deep state-space encoders based on the concept of state reconstructability and an efficient multiple-shooting formulation of the squared loss of the prediction error to estimate the dynamics and the lifted state only from input-output data. Furthermore, the Koopman model structure includes an innovation noise term that is used to handle process and measurement noise. It is shown that the proposed approach is statistically consistent (estimation error tends to zero when the number of data points goes to infinity) and computationally efficient due to the multiple-shooting formulation, by which the prediction error of the model can be calculated on multiple subsections of the data in parallel. The latter allows for efficient batch optimization of the network parameters and, at the same time, excellent long-term prediction capabilities of the obtained models. The performance of the approach is illustrated by nonlinear benchmark examples and experimental data from a Crazyflie 2.1 quadcopter.

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.

Referee Report

2 major / 2 minor

Summary. The paper presents a method for identifying Koopman models of nonlinear systems with inputs from noisy input-output data. It uses deep state-space encoders based on state reconstructability, augments the model with an innovation noise term to handle process and measurement disturbances, and employs a multiple-shooting formulation of the squared prediction-error loss. The central claims are statistical consistency (estimation error tends to zero as the number of data points N goes to infinity) and computational efficiency due to parallelizable multiple shooting, with demonstrated long-term prediction performance on nonlinear benchmarks and Crazyflie 2.1 quadcopter experiments.

Significance. If the consistency result is rigorously established, this would advance data-driven modeling and control of nonlinear systems in realistic noisy settings by extending Koopman operator methods with deep encoders and efficient optimization. The multiple-shooting approach for scalable batch optimization and long-term prediction is a clear strength. Experimental validation on hardware data adds practical value.

major comments (2)
  1. [Theoretical analysis / consistency result] The statistical consistency claim (estimation error → 0 as N → ∞) is load-bearing but rests on the deep encoders recovering an exact lifted representation under the reconstructability property while the innovation term absorbs all disturbances. For general (non-white, possibly state-dependent) noise, reconstructability from I/O data alone is not guaranteed for arbitrary nonlinear systems, and no mechanism enforces it; residual reconstruction error would prevent convergence of the empirical minimizer to true parameters. A detailed proof sketch, error bounds, or additional identifiability conditions are needed to support this.
  2. [Method / multiple-shooting formulation] The multiple-shooting formulation is credited with both efficiency and excellent long-term prediction, but it is unclear how the parallel subsection loss interacts with the innovation noise term to preserve consistency. The abstract states the loss is computed on multiple subsections in parallel; a specific derivation showing that this does not introduce bias or violate the asymptotic properties would clarify the central efficiency-consistency tradeoff.
minor comments (2)
  1. [Abstract] The abstract refers to 'rather general noise conditions' without specifying the precise class (e.g., bounded moments, independence assumptions); a short clarifying sentence would improve precision.
  2. [Method] Notation for the lifted state and innovation term should be introduced with explicit definitions early in the method section to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback on our manuscript. We address each major comment below with clarifications on our approach and indicate planned revisions to strengthen the presentation.

read point-by-point responses

  1. Referee: [Theoretical analysis / consistency result] The statistical consistency claim (estimation error → 0 as N → ∞) is load-bearing but rests on the deep encoders recovering an exact lifted representation under the reconstructability property while the innovation term absorbs all disturbances. For general (non-white, possibly state-dependent) noise, reconstructability from I/O data alone is not guaranteed for arbitrary nonlinear systems, and no mechanism enforces it; residual reconstruction error would prevent convergence of the empirical minimizer to true parameters. A detailed proof sketch, error bounds, or additional identifiability conditions are needed to support this.

    Authors: We appreciate the referee's emphasis on rigorously supporting the consistency result. The method relies on deep encoders designed around the state reconstructability property to recover a lifted representation from input-output data, with the innovation term explicitly modeling general disturbances (including non-white and state-dependent noise) so that the prediction-error objective remains well-defined. While the manuscript establishes consistency under these modeling assumptions as N tends to infinity, we agree that the current presentation would benefit from greater detail. In the revision we will add an appendix containing a proof sketch that outlines the key steps: (i) uniform approximation of the reconstructability map by the deep network class, (ii) convergence of the empirical minimizer of the innovation-augmented loss to the population risk, and (iii) identifiability of the Koopman parameters once the lifted state is recovered. We will also state the precise technical conditions (e.g., persistence of excitation and network capacity) under which the result holds. revision: yes

  2. Referee: [Method / multiple-shooting formulation] The multiple-shooting formulation is credited with both efficiency and excellent long-term prediction, but it is unclear how the parallel subsection loss interacts with the innovation noise term to preserve consistency. The abstract states the loss is computed on multiple subsections in parallel; a specific derivation showing that this does not introduce bias or violate the asymptotic properties would clarify the central efficiency-consistency tradeoff.

    Authors: The multiple-shooting loss is formed by partitioning the data into contiguous subsections and summing the squared prediction errors (each computed with its own innovation sequence) across all subsections. Because the subsections are non-overlapping and together exhaust the full dataset, the total objective is mathematically identical to the single-shooting prediction-error loss; the innovation term is simply applied segment-wise. Consequently, the empirical risk minimizer and its asymptotic convergence properties remain unchanged. In the revised manuscript we will include a short derivation in the main text (or appendix) that explicitly shows the equivalence of the summed subsection losses to the full-trajectory loss and confirms that the parallel formulation introduces neither bias nor alteration to the consistency argument. revision: yes

Circularity Check

0 steps flagged

No significant circularity: consistency claim rests on explicit assumptions and multiple-shooting loss without reduction to fitted inputs

full rationale

The paper states that statistical consistency holds under the state reconstructability property and with an added innovation noise term that absorbs disturbances. The multiple-shooting formulation is introduced solely for computational efficiency in parallel loss evaluation on data subsections; it does not define or force the consistency result. No equations or self-citations in the provided abstract reduce the consistency claim to a tautology or to parameters fitted from the target quantity itself. The derivation therefore remains self-contained against external benchmarks once the reconstructability assumption is granted.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of a reconstructable state representation and on the modeling assumption that an innovation noise term suffices for general noise; no explicit free parameters beyond network weights are declared in the abstract.

axioms (2)
  • domain assumption The nonlinear system admits a state reconstructability property that allows recovery of the lifted Koopman state from input-output trajectories.
    Invoked to justify the deep state-space encoder architecture.
  • domain assumption An additive innovation noise term inside the Koopman model structure is sufficient to capture both process and measurement disturbances.
    Used to handle general noise conditions without further statistical modeling.

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