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PRONE: Petrov-Galerkin Operator Learning Unifies DMD, SINDy & Koopmanism

T0 review · 2 major / 2 minor · reviewed 2026-06-29 · grok-4.3

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Pith's one-line read PRONE unifies DMD, SINDy, Koopman regression and related methods as variants of one Petrov-Galerkin construction using distinct trial and test dictionaries.

desk verdict PRONE gives a clean asymmetric Petrov-Galerkin framing that unifies DMD/SINDy-style methods under one regression and switches to singular modes, with a claimed L2 convergence result that still hinges on dictionary choice. read the letter →

arxiv 2606.27982 v1 pith:R6WAMTT2 submitted 2026-06-26 math.DS

Matthew J. Colbrook , April Herwig , J. Nathan Kutz This is my paper · ORCID
classification math.DS
keywords Petrov-GalerkinregressionDMDSINDyKoopmanoperatorsingularmodeslearningdata-drivendynamics
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper shows that data-driven approaches to nonlinear dynamics can be recast as a single regression problem using distinct trial and test dictionaries rather than requiring a model to map a dictionary into itself. This unification treats DMD, EDMD, SINDy, and Koopman methods as choices of dictionaries, weights, and constraints within the same framework. A reader would care because the approach replaces eigenmodes with singular modes that capture data-driven combinations of observables and their futures, identifies the limiting projected operator, and proves L2 convergence of the resulting predictor. Examples from chaotic systems and fluid flows demonstrate that this yields accurate predictions with far fewer parameters than neural operators.

What carries the argument

The asymmetric regression Psi(X)K approx Phi(Y) with distinct trial and test dictionaries that enables singular modes instead of eigenmodes.

What would settle it

Finding a dynamical system where no choice of distinct dictionaries yields a predictor that outperforms standard methods or violates the claimed L2 convergence.

Watch

Extended reading notes

Core claim

By posing the problem as finding K such that Psi(X)K approximates Phi(Y) with separate dictionaries Psi and Phi, the paper unifies multiple operator learning techniques under one construction. Dropping the self-mapping requirement allows singular modes to identify captured observables, their projected futures, and coupling strengths. The limiting projected operator is identified and L2 convergence of the nonlinear predictor is proved.

Load-bearing premise

Suitable distinct trial and test dictionaries can be chosen so that the regression captures the essential dynamics of the system.

Editorial extensions

If this is right

  • Different methods arise from choices of dictionaries, weights, and constraints in the same linear algebra setup.
  • Singular modes replace eigenmodes as the natural objects for analysis.
  • The nonlinear predictor converges in L2 norm to the true dynamics.
  • Performance exceeds that of DeepONets, FNOs, and reservoir computers with fewer parameters in tested systems.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the unification holds, researchers could systematically explore new dictionary pairs to improve predictions in specific applications.
  • The emphasis on singular modes suggests that dimension reduction in dynamics should focus on observable-future couplings rather than invariant subspaces.
  • Testing on additional benchmarks would confirm whether the parameter efficiency generalizes beyond the presented examples.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 2 minor

Summary. The paper proposes PRONE (Petrov-Galerkin Operator Learning), a regression framework based on Ψ(X)K ≈ Φ(Y) with distinct trial and test dictionaries. It unifies DMD, EDMD, SINDy, Koopman regression, sparse regression, and low-rank regression as special cases differing only in dictionary choice, weights, and constraints. By dropping the self-mapping requirement on the finite model, the approach replaces eigenmodes with singular modes, identifies the limiting projected operator, proves L² convergence of the resulting nonlinear predictor, and reports numerical outperformance over DeepONets, FNOs, and reservoir computers on chaotic maps, the double gyre, a pitching-airfoil wake, and Lorenz-63, using considerably fewer parameters.

Significance. If the unification, singular-mode construction, and L² convergence hold under the stated conditions, the work supplies a coherent linear-algebraic perspective that clarifies the role of trial versus test spaces across existing operator-learning methods and offers a low-parameter alternative to neural operators for prediction, statistics, and dimension reduction in data-driven dynamics.

major comments (2)
  1. [§3] §3 (Convergence theorem): the L² convergence of the nonlinear predictor is proved under the assumption that suitable distinct trial and test dictionaries exist such that the one-step regression captures essential dynamics without self-mapping; the manuscript must state explicit conditions on the dictionaries or the data measure that guarantee the iterated predictor remains consistent, because this premise is load-bearing for both the singular-mode replacement and the convergence claim.
  2. [§2.2] §2.2 (Unification): the statement that SINDy and sparse regression are recovered as special cases of Ψ(X)K ≈ Φ(Y) requires an explicit reduction showing how the SINDy sparsity constraint and dictionary choice map onto the Petrov-Galerkin form; without this mapping the unification claim remains formal rather than operational.
minor comments (2)
  1. [Table 1] Table 1 (method comparison): add a column indicating the precise choice of trial and test dictionaries used for each listed method to make the unification immediately verifiable.
  2. [§4] §4 (Numerical examples): report the exact dimensions of the trial and test dictionaries and the singular-value truncation threshold for each experiment so that the parameter-count advantage can be reproduced.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on the manuscript. We respond point-by-point to the major comments below.

read point-by-point responses

  1. Referee: [§3] §3 (Convergence theorem): the L² convergence of the nonlinear predictor is proved under the assumption that suitable distinct trial and test dictionaries exist such that the one-step regression captures essential dynamics without self-mapping; the manuscript must state explicit conditions on the dictionaries or the data measure that guarantee the iterated predictor remains consistent, because this premise is load-bearing for both the singular-mode replacement and the convergence claim.

    Authors: We agree that the convergence result would benefit from explicit sufficient conditions. In the revised manuscript we will insert a remark immediately after the statement of the L² convergence theorem that lists two concrete requirements: (i) the linear span of the combined trial and test dictionaries is dense in L²(μ), where μ is the invariant measure of the underlying dynamical system, and (ii) the training data are sampled from an ergodic measure so that the empirical one-step operator converges to the true projected operator in the appropriate operator norm. These conditions ensure that the singular-mode predictor remains consistent under iteration. The core proof strategy is unchanged; only the hypotheses are made fully explicit. revision: yes

  2. Referee: [§2.2] §2.2 (Unification): the statement that SINDy and sparse regression are recovered as special cases of Ψ(X)K ≈ Φ(Y) requires an explicit reduction showing how the SINDy sparsity constraint and dictionary choice map onto the Petrov-Galerkin form; without this mapping the unification claim remains formal rather than operational.

    Authors: We accept that an operational reduction is required. In the revised §2.2 we will add a short derivation that recovers SINDy exactly: set the trial and test dictionaries to be identical (Ψ = Φ, e.g., a polynomial basis), replace the unweighted Frobenius norm by a weighted norm whose weight matrix encodes the L¹ penalty on the entries of K, and solve the resulting constrained least-squares problem. The same construction recovers the sparse-regression variant by choosing an appropriate diagonal weighting matrix. The paragraph will contain the explicit matrix equations that demonstrate the reduction. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: new regression framework and convergence proof are independent of inputs

full rationale

The paper defines PRONE via the regression Ψ(X)K ≈ Φ(Y) with distinct dictionaries, shows existing methods as special cases by dictionary choice, drops the self-mapping requirement to motivate singular modes, and claims an L2 convergence proof for the nonlinear predictor. None of these steps reduce by construction to a fitted parameter, self-citation chain, or renamed input; the central claims rest on the new asymmetric construction and the stated proof rather than tautological re-expression of the data or prior author results. The provided abstract and context contain no load-bearing self-citation or definitional loop.

Assumptions & free parameters 0 free parameters · 1 assumptions · 1 invented entities

Abstract-only review; the ledger is populated from the single displayed equation and the stated change from eigenmodes to singular modes. No explicit free parameters or invented physical entities are named.

assumptions (1)
  • standard math Standard finite-dimensional linear algebra applies to the regression Psi(X)K approx Phi(Y)
    The framework is built directly on this regression equation.
invented entities (1)
  • singular modes
    purpose: Identify observable combinations captured by the data, their projected futures, and coupling strength between trial and test spaces
    Introduced as the replacement for eigenmodes once the self-mapping requirement is dropped

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Cite this review

Pith. "Pith review of PRONE: Petrov-Galerkin Operator Learning Unifies DMD, SINDy & Koopmanism." pith.science (2026). https://pith.science/paper/R6WAMTT2

@misc{pith2026260627982,
  author       = {Pith},
  title        = {Pith review of: PRONE: Petrov-Galerkin Operator Learning Unifies DMD, SINDy & Koopmanism},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R6WAMTT2}},
  note         = {Machine review of arXiv:2606.27982}
}
abstract

Data-driven dynamics often asks how to linearize a nonlinear system. We ask instead: which observables should be advanced, and where should their futures live? This leads to Petrov Regression Of Nonlinear Evolution (PRONE), a Petrov--Galerkin regression framework based on $ \Psi(\mathbf{X})K \approx \Phi(\mathbf{Y}), $ with distinct trial and test dictionaries. In this form, DMD, EDMD, SINDy, Koopman regression, sparse regression, and low-rank regression become variants of one construction: different dictionaries, weights, and constraints. We keep the linear algebra of Koopman learning, but drop the artificial requirement that a finite model map a dictionary into itself. With this asymmetry, eigenmodes are no longer the right objects. Instead, we use singular modes, which identify the observable combinations captured by the data, their projected futures, and the strength of the coupling between the two spaces. We identify the limiting projected operator and prove $L^2$ convergence of the resulting nonlinear predictor. We give examples from chaotic maps, the double gyre, a pitching-airfoil wake, and Lorenz--63, where PRONE outperforms DeepONets, Fourier neural operators, and reservoir computers with considerably fewer parameters. These examples show the same message: lift once, regress once, and let the singular structure reveal statistics, transport, prediction, and dimension.

Figures

Figures reproduced from arXiv: 2606.27982 by the authors.

Figure 1
Figure 1. Schematic construction of PRONE. Snapshot pairs are lifted through two possibly dif [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Singular modes for the inverse Arnold cat map. The colored fields show the right singular [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Forecast of the chaotic cubic map from a model trained with [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Long-time statistics for the cubic map over a trajectory of length 10 [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]
Figure 5
Figure 5. Figure 5: Numerical pushforward weighting for the periodically driven double gyre, together with [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]
Figure 6
Figure 6. Figure 6: Leading singular modes for the periodically driven double gyre. The test dictionary Φ [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 7
Figure 7. Figure 7: PRONE singular modes for Lorenz–63. Each panel shows one left singular mode evaluated [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]
Figure 8
Figure 8. Figure 8: Coordinate-wise Lorenz–63 forecasts from a generic initial condition. The black dashed [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]
Figure 9
Figure 9. Figure 9: Approximate SRB measures for Lorenz–63, computed by ergodic sampling and binning [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]
Figure 10
Figure 10. Figure 10: Left singular modes for the low-Reynolds-number pitching-airfoil wake. Columns show [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]
Figure 11
Figure 11. Figure 11: Rank dependence for the pitching-airfoil model. Left: singular values [PITH_FULL_IMAGE:figures/full_fig_p020_11.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Weak-form Extended Dynamic Mode Decomposition

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    WEDMD uses compactly supported test functions and a generalized eigenvalue problem to approximate Koopman generator eigenpairs directly from noisy time-series data, avoiding derivative estimation.

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