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arxiv: 2605.31438 · v1 · pith:5IJGN5FKnew · submitted 2026-05-29 · 💻 cs.LG

Flow map learning in nonlinear vector autoregressive models: influence of the feature-library structure on the training error

Markus Gross This is my paper

Pith reviewed 2026-06-28 23:17 UTC · model grok-4.3

classification 💻 cs.LG
keywords nonlinear vector autoregressive modelsflow map learningtraining error scalingfeature libraryLie-series coefficientsNVARNG-RCdynamical systems
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The pith

The training error of nonlinear vector autoregressive models scales with time resolution according to laws set by whether the feature library exactly represents the early coefficients in the flow map expansion.

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

This paper investigates the identifiability of learning nonlinear dynamical systems using NVAR models. It establishes that training error versus time resolution exhibits specific scaling behaviors. These behaviors hinge on the feature library's ability to match the initial Lie-series terms of the flow map exactly or only approximately. Demonstrations use polynomial-governed systems with monomial and Fourier libraries. The analysis covers effects of resolution, nonlinearity degree, and delay terms on error and forecast quality.

Core claim

The training error as a function of time resolution follows characteristic (pre-)asymptotic scaling laws that depend on whether the feature library can represent the early Lie-series coefficients of the flow map exactly or merely approximately. For dynamical systems governed by polynomial vector fields, this is shown for monomial and Fourier feature libraries in NVAR models, revealing the dependence on temporal resolution, nonlinear degree, and number of delay terms.

What carries the argument

The feature library structure and its exact or approximate representation of early Lie-series coefficients of the flow map, which governs the scaling laws of the training error.

Load-bearing premise

The dynamical systems under consideration are governed by polynomial vector fields.

What would settle it

An experiment that computes training errors for successively finer time resolutions in a polynomial vector field system and checks whether the error follows the predicted scaling for exact versus approximate library representations.

Figures

Figures reproduced from arXiv: 2605.31438 by Markus Gross.

Figure 1
Figure 1. Figure 1: Effect of increasing the monomial degree [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) Dependence of the training error, forecast horizon, and the condition number (of the training [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (a,c,e) Dependence of the training error and forecast horizon (in units of the Lyapunov time) [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Semigroup consistency test. The semigroup defect [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Effect of delay coordinates on the training error and forecast horizon for the Halvorsen model. [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: A linear delay predictor learns a harmonic approximation (solid curve) of a (nonlinear) time series [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Dependence of the training error ε/ε0 and the forecast horizon Tfch (in units of the Lyapunov time TL) on the number of time taps Γ and the monomial degree M of the feature map for various dynamical systems. The theoretical prediction (3.19) (dashed line) gives an asymptotic error scaling for infinite resolution ∆t → 0, neglecting Γ-dependent prefactors. In our convention, Γ = 1 corresponds to purely insta… view at source ↗
Figure 8
Figure 8. Figure 8: Comparison between readout weights (a,c,e) of a NG-RC model trained on the Halvorsen system [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: NG-RC with a Fourier feature library trained on various polynomial dynamical systems. The left [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Semigroup consistency test for a NG-RC with Fourier-features. The semigroup defect [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Comparison of the learned effective flow map of a NG-RC with Fourier features (a,c) with the [PITH_FULL_IMAGE:figures/full_fig_p018_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Theoretically predicted training error εtrain ≃ a∆t r ∗+1 [Eq. (3.8)] and forecast horizon Tfch ≃ ln(C/εtrain) [Eq. (C12)] as functions of ∆t and M. We use values for the fitting parameters a and C ≡ εtol/c obtained from numerical experiments. [2] W. Gilpin, Model scale versus domain knowledge in statistical forecasting of chaotic systems, Phys. Rev. Research 5, 043252 (2023). [3] M. Yan, C. Huang, P. Bie… view at source ↗
read the original abstract

Time series forecasting often requires learning nonlinear and time-delayed dependencies. A paradigmatic class of forecasting models are nonlinear vector autoregressive processes (NVAR), also known as next-generation reservoir computers (NG-RCs). These models approximate the Koopman operator on the space spanned by their explicit feature library. We consider the identifiability problem for learning Markovian nonlinear dynamical systems and show that the training error as a function of time resolution follows characteristic (pre-)asymptotic scaling laws. These laws depend on whether the feature library can represent the early Lie-series coefficients of the flow map (propagator) exactly or merely approximately. For dynamical systems governed by polynomial vector fields, we demonstrate the mechanism for NVAR/NG-RC models with monomial and Fourier feature libraries. We determine the dependence of the training error on the temporal resolution, the involved nonlinear degree, and the number of delay terms. While delay terms reduce the optimal one-step training error, they improve long-horizon forecasts only when the library provides sufficient nonlinearity. Thus, small training error coexists with weak generalization as the model class is mismatched to the true data-generating process. Numerical experiments on various chaotic dynamical systems confirm the theoretical predictions.

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

0 major / 3 minor

Summary. The paper examines the identifiability problem in learning Markovian nonlinear dynamical systems via nonlinear vector autoregressive (NVAR/NG-RC) models, which approximate the Koopman operator using an explicit feature library. It derives that the training error as a function of time resolution obeys characteristic (pre-)asymptotic scaling laws determined by whether the library exactly or approximately represents the early Lie-series coefficients of the flow map. The analysis is restricted to polynomial vector fields and demonstrated explicitly for monomial and Fourier libraries, including the dependence of error on temporal resolution, nonlinear degree, and delay count. The work also shows that delay terms reduce optimal one-step training error but improve long-horizon forecasts only with sufficient nonlinearity in the library, and confirms the scaling predictions via numerical experiments on chaotic systems.

Significance. If the scaling laws and their dependence on exact versus approximate representation hold, the result supplies a concrete mechanism explaining why small training error can coexist with poor generalization when the model class mismatches the data-generating process. It clarifies the interplay between temporal resolution, library structure, and delay embedding in these Koopman-based forecasters, offering practical guidance for library design. The restriction to polynomial vector fields with explicit Lie-series termination, combined with numerical confirmation on standard chaotic benchmarks, constitutes a clear, falsifiable contribution within its stated scope.

minor comments (3)
  1. The abstract and introduction state the restriction to polynomial vector fields clearly, but §2 or the methods section should include an explicit remark that the exact-vs-approximate distinction and resulting scalings do not automatically extend to non-polynomial fields where the Lie series is infinite; this would prevent misreading by readers focused on general nonlinear systems.
  2. Figure captions for the numerical experiments should state the precise values of time step Δt, library degree, and number of delays used in each panel so that the reported scaling exponents can be directly compared to the derived expressions without consulting the main text.
  3. Notation for the Lie-series coefficients and the feature-library projection operator is introduced without a consolidated table; adding a short notation table would improve readability when the scaling arguments are revisited in the discussion of delay terms.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their detailed summary of our manuscript and for the positive assessment of its contribution. The recommendation for minor revision is noted. No specific major comments were provided in the report, so we have no points to address point-by-point. We are prepared to incorporate any minor editorial changes requested by the editor.

Circularity Check

0 steps flagged

Derivation of scaling laws is self-contained; no circular reduction

full rationale

The paper derives (pre-)asymptotic scaling laws for training error versus time resolution from the structure of Lie-series coefficients of the flow map and whether monomial/Fourier libraries represent them exactly or approximately, restricted to polynomial vector fields. This is a first-principles analysis of model mismatch in NVAR/NG-RC setups, not a fit to data followed by a renamed prediction, nor a self-definitional loop, nor load-bearing self-citation. The abstract and description provide no equations or steps that reduce the claimed laws to inputs by construction; the result is independent of fitted parameters and externally falsifiable via the polynomial assumption and Lie expansion. No circularity patterns are exhibited.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no explicit free parameters, axioms, or invented entities are detailed beyond the stated modeling assumptions.

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Reference graph

Works this paper leans on

126 extracted references · 41 canonical work pages · 9 internal anchors

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    For raw monomial libraries, the RMS training error behaves approximately asε∼a(∆t) bM +ε floor(M, λ), with constantsa, b

    Effect of monomial degree Figure 1 illustrates the training error and forecast horizon (obtained for the Halvorsen model) in dependence of the monomial degreeMof the feature library and the Ridge regularizationλ. For raw monomial libraries, the RMS training error behaves approximately asε∼a(∆t) bM +ε floor(M, λ), with constantsa, b. The first term is the ...

  2. [2]

    (3.8) until a floor is reached

    Effect of temporal resolution Figure 2(a,b) shows in more detail the behavior of the training error, following closely Eq. (3.8) until a floor is reached. The forecast horizonT fch grows according to Eq. (C12) with decreasingε. The saturation of the training error and forecast horizon is due to the integrator accuracy used to generate the trajectories and...

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    When the model can reproduce the early Lie derivatives of the true flow map, the error in the approximation in Eq

    Semigroup consistency The semigroup consistency of a learned flow map bΦ∆t requires bΦℓ∆t ≈bΦℓ ∆t (4.7) wherebΦℓ ∆t =bΦ∆t ◦bΦ∆t ◦· · ·◦bΦ∆t (ℓtimes). When the model can reproduce the early Lie derivatives of the true flow map, the error in the approximation in Eq. (4.7) is proportional to the training error [see Eq. (A55)], Sm :=∥bΦℓ∆t −bΦℓ ∆t∥ ∼ε train.(...

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    Effect of delay terms Figure 5 demonstrates the influence of the number of time taps Γ on training error and forecast horizon at fixed monomial degreeM(Halvorsen model with ∆t= 0.01). When using only linear delays (M= 1), the NG-RC model becomes a linear autoregressive process and learns a harmonic approximation, often with a slowly decaying amplitude env...

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    8, we compare the learned readout weights of a NG-RC trained on the Halvorsen system to the theoretical predictions in Section III B

    Readout weights In Fig. 8, we compare the learned readout weights of a NG-RC trained on the Halvorsen system to the theoretical predictions in Section III B. A feature map consisting only of linear delay terms results in the generic weights of Eq. (3.17) (panels a,b). Conversely, if the feature map consists only of instantaneous monomials (panels c,d), th...

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    For a fixed target functionh∈L 2(µ;R d), the projectionPhis the best approximation tohamong all functions in V

    A library-agnostic projection framework Given the feature mapg, the associated model class consists of all vector-valued functions that can be represented by a linear readout, V:= Hg(·) :H∈R d×n ⊂L 2(µ;R d).(A3) HereHis only a generic coefficient matrix parameterizing the elements ofV. For a fixed target functionh∈L 2(µ;R d), the projectionPhis the best a...

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    Asymptotic versus pre-asymptotic regimes If the vector field is not exactly representable by the feature library,f/∈V, thenε 1 >0 and the true asymptotic law is necessarily E(∆t) =ε 2 1∆t2 +O(∆t 3),(A14) Ifε 1 is much smaller thanε m for somem >1, different pre-asymptotic scalings are possible. Then, over a window of ∆tin which them-th term dominates in (...

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    Polynomial vector fields In order to make the previous discussion more concrete, consider the case whenfis polynomial of total degreep. In that case each Lie coefficientF m =L m−1 f fis again polynomial, with the degree bound degF m ≤1 +m(p−1), p >1.(A17) Hence the question reduces to how well the library reproduces polynomials of the degrees that are pro...

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    , N} d ,(A22) with base frequencyω 0

    Fourier feature libraries We now consider flow map approximation with aN-frequency Fourier feature library: g(N)(x) = sin ω0 k·x ,cos ω0 k·x :k∈ {0, . . . , N} d ,(A22) with base frequencyω 0. Such a library can show extended pre-asymptotic error scaling before crossing over to the asymptotic law∝∆t 2. We assume the data reside in a box Ω⊂[−R, R] d, and d...

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    Connection to bias–variance decomposition The projection error Eq. (A7) is the approximation-bias component of the usual bias–variance decomposition. In the increment-prediction setting of Section A 1, the target isF ∆t ∈L 2(µ;R d) and the model class is V={Hg(·) :H∈R d×n}.(A41) Leth ⋆ :=PF ∆t denote theL 2(µ)-orthogonal projection ofF ∆t ontoV. Then, for...

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    Consider now the difference betweenF [r] ∆t and the approximation of the true flow map by the feature library,PF ∆t ≈ ˆHg: in the same approximation-dominated regime as in Eq

    Truncated flow-map error Since the exact expression for the flow mapF∆t is usually not available, consider its approximation to orderr F[r] ∆t(x) := Φ[r] ∆t(x)−x= rX m=1 ∆tm m! Fm(x),(A42) and define the difference to the true flow map asR [r] ∆t :=F ∆t −F [r] ∆t. Consider now the difference betweenF [r] ∆t and the approximation of the true flow map by th...

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    In the population limit, bΦh(x) :=x+PF h(x),(A51) theℓ-step semigroup defect is given by Sℓ(h)2 := bΦℓh −bΦℓ h 2 L2(µ) ,(A52) wherebΦℓ h denotesℓ-fold composition

    Semigroup consistency The exact flow map satisfies the semigroup property [29] Φs+t = Φs ◦Φ t.(A50) This can be tested for the learned flow map bΦh, and we show that it essentially follows the same ∆t-scaling as the training error. In the population limit, bΦh(x) :=x+PF h(x),(A51) theℓ-step semigroup defect is given by Sℓ(h)2 := bΦℓh −bΦℓ h 2 L2(µ) ,(A52)...

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    Error recurrence in autonomous forecasting We study the error evolution during autonomous forecasting. Letx n+1 = Φ∆t(xn) be the true dynamics and ˆxn+1 = ˆH g(ˆxn) be the autonomous prediction obtained from the trained model. Denoting the deviation between the two bye n =∥x n −ˆxn∥, we have en+1 =∥Φ ∆t(xn)− ˆH g(ˆxn)∥ ≤ ∥Φ ∆t(xn)−Φ ∆t(ˆxn)∥+∥Φ ∆t(ˆxn)− ˆ...

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