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arxiv: 2606.06728 · v2 · pith:236KYGU2new · submitted 2026-06-04 · 🧮 math.DS

Data-driven methods for computation of optimal linear response in high-dimensional dynamical systems

Gary Froyland , Dimitrios Giannakis , Nicholas Peters This is my paper

Pith reviewed 2026-06-27 23:01 UTC · model grok-4.3

classification 🧮 math.DS
keywords optimal linear responsetransfer operatorsKoopman operatorsdynamical systemsdata-driven methodsspectrum manipulationalmost-invariant setshigh-dimensional systems
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The pith

Kernel-smoothed approximations of transfer operators enable data-driven optimization of infinitesimal perturbations to manipulate system spectra.

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

The paper develops a method to identify the smallest change to a nonlinear dynamical system that produces a chosen effect on its long-term statistics, such as slowing the loss of memory in almost-cycles or shifting their oscillation rates. It builds kernel-smoothed estimates of the transfer and Koopman operators directly from trajectory data, even when observations are high-dimensional, then uses these estimates inside an optimization problem to locate the desired perturbation. The same construction yields vector fields that show how the perturbation alters the observed behavior. Applications to periodic, chaotic, and climate models produce perturbations that align with the stated goals.

Core claim

Kernel-smoothed approximations of the transfer and Koopman operators, built from trajectory data, are inserted into a spectral optimization problem whose solution is the optimal infinitesimal perturbation that realizes any prescribed manipulation of the spectrum, including increases in frequency or reductions in decay rate for eigenvalues tied to almost-cycles or almost-invariant sets.

What carries the argument

Kernel-smoothed approximations of the transfer and Koopman operators that turn spectral manipulation into a tractable optimization problem solved from data.

If this is right

  • The optimization can be solved to increase frequency or suppress decay of correlations for almost-cycles identified by the approximated operator.
  • Optimal-response vector fields constructed from the same data visualize the physical effect of the perturbation under any chosen observations.
  • The procedure scales to high-dimensional trajectory data, as shown by its application to an Earth system model for El Nino Southern Oscillation.
  • The resulting perturbations are nontrivial yet consistent with the target dynamical objectives in the tested periodic, chaotic, and climate examples.

Where Pith is reading between the lines

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

  • If the transfer from approximation to true system holds, the same workflow could locate targeted adjustments that lengthen or shorten specific climate oscillations.
  • The vector-field construction might be reused to interpret how a chosen perturbation affects statistics under observations different from those used to build the operator.
  • Repeating the procedure on systems whose exact linear response is known analytically would directly test how much approximation error is tolerable before the computed perturbation loses its optimality.

Load-bearing premise

The kernel-smoothed operator approximations remain accurate enough that perturbations optimal for the approximation remain optimal for the true nonlinear system.

What would settle it

Take the perturbation found from the approximated operators, apply it to the original dynamical system, and measure whether the relevant eigenvalues of the true transfer operator move exactly as the optimization predicted.

Figures

Figures reproduced from arXiv: 2606.06728 by Dimitrios Giannakis, Gary Froyland, Nicholas Peters.

Figure 1
Figure 1. Figure 1: Left: Heat map of a 50 × 50 diagonal block of the 1000 × 1000 transition matrix P0. Right: Spectrum of P0, visualized in terms of the magnitudes, |λ (k) |, and frequencies ν (±k) = arg λ (±k)/2π∆t, associated with the eigenvalues λ (±k) . ν (±k) = arg λ (±k)/2π ∆t, are displayed in [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Left: Heat map of a 50×50 diagonal block of the 1000×1000 perturbation matrix P˙ . Right: The optimal-response vector field (red arrows), for the frequency￾increasing perturbation, originating on the corresponding observational data points. The circle rotation direction is anticlockwise. occurs on the geometric circle in R 2 . We emphasise that there is no a priori con￾straint on the optimal-response vecto… view at source ↗
Figure 3
Figure 3. Figure 3: Left: The variable rotation angle α(θ) (black line) plotted as a function of the angle θ. Also plotted are the magnitudes of the orthogonal projections of the optimal-response vector fields onto the tangent spaces of the instantaneous velocities in R 2 , for the frequency-increasing perturbation (red) and magnitude-increasing per￾turbation (green). Centre: Magnitudes and frequencies of the eigenvalues of P… view at source ↗
Figure 4
Figure 4. Figure 4: Left: The observational data is coloured according to the forward differ￾ence α(θ). The arrows represent the optimal-response vector field of the frequency￾increasing perturbation at a selection of points on the circle. Right: The same infor￾mation for the magnitude-increasing perturbation. The scaling factor and number of arrows were chosen independently to aid in visualisation, and are not the same betwe… view at source ↗
Figure 5
Figure 5. Figure 5: Magnitudes and frequencies of the leading eigenvalues of [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Left: The observational data is coloured according to the argument of v0. The arrows represent the optimal-response vector field of the frequency perturba￾tion. For visibility, arrow lengths have been uniformly scaled and only a random sample of arrows are drawn, sampled with weighting inversely proportional to the density of points. Right: Entries of v0 are plotted in the complex plane, coloured by their … view at source ↗
Figure 7
Figure 7. Figure 7: Left: The observational data is coloured according to the magnitude of v0. The arrows represent the optimal-response vector field of the magnitude perturbation. Arrow selection and length scaling are as in [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The magnitudes and frequencies of the leading eigenvalues of [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The observational data is coloured according to the eigenvector [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Left: The magnitudes and frequencies of the largest 30 eigenvalues of P0, approximating those of the transfer operator for the SST data. Right: The real and imaginary parts of the ENSO eigenvector. The magnitude and angle of each point can be interpreted respectively as the ENSO strength and phase during each month of the data. The cycle is broken up into 8 phases (polar sectors) each spanning π/4 radians… view at source ↗
Figure 11
Figure 11. Figure 11: Left: Phase composites of SST anomalies (colours) and surface wind anomalies (arrows) based on the ENSO eigenvector v0. Each arrow represents the mean wind anomaly in the surrounding 7.5 ◦×7.5 ◦ square. Center: Phase composites of the optimal-response vector fields for the SST field (colours) and surface wind field (arrows), for the ENSO frequency-increasing perturbation. Right: The same information for t… view at source ↗
Figure 12
Figure 12. Figure 12: SST anomalies (black line) and optimal-response SST fields for the [PITH_FULL_IMAGE:figures/full_fig_p029_12.png] view at source ↗
read the original abstract

We develop a data-driven framework for estimating optimal linear response of nonlinear dynamical systems. Our approach is based on kernel-smoothed approximations of the transfer/Koopman operators of the system, built from possibly high-dimensional observations along trajectories. Combining these operator approximations with the theory developed in [Antown et al. (2018), J. Stat. Phys., 170(6), 1051-1087], we formulate a computationally tractable optimization problem for the optimal infinitesimal perturbation realising any desired manipulation of the spectrum. We also introduce a notion of optimal-response vector fields for visualising, and physically interpreting, the system's response to the optimal perturbation under arbitrary observations. Our focus is on finding perturbations that optimally increase the frequency or optimally suppress the decay of correlations of almost-cycles or almost-invariant sets associated with the eigenvalues of the kernel-smoothed transfer operator. We illustrate our approach with applications to low-dimensional periodic and chaotic systems, as well as a high-dimensional example involving the El Nino Southern Oscillation in a comprehensive Earth system model. In these examples our approach discovers nontrivial optimal perturbations of the system, which are post hoc natural and consistent with the desired dynamical objectives.

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

3 major / 2 minor

Summary. The paper develops a data-driven framework for estimating optimal linear response in nonlinear dynamical systems. It uses kernel-smoothed approximations of transfer/Koopman operators constructed from trajectory data (possibly high-dimensional), combines these with the optimization theory of Antown et al. (2018) to pose a tractable problem for infinitesimal perturbations that achieve desired spectral manipulations (e.g., increasing frequency or suppressing correlation decay of almost-cycles), and introduces optimal-response vector fields for interpretation. The method is illustrated on low-dimensional periodic/chaotic systems and a high-dimensional ENSO example in an Earth system model, where the discovered perturbations are reported as nontrivial yet physically natural.

Significance. If the kernel approximations are shown to yield perturbations whose spectral effects transfer to the true nonlinear operator, the framework would offer a practical, observation-based route to optimal control of spectral properties in high-dimensional systems, with direct relevance to climate dynamics and chaos control. The combination of data-driven operator learning with an external optimization theory is a natural extension, and the introduction of optimal-response vector fields provides a useful interpretive tool.

major comments (3)
  1. [Formulation of the optimization problem] Formulation of the optimization problem (section referenced in the abstract): the kernel-smoothed operator is used to define the optimization problem, but no a-priori error bound or Lipschitz-type control is given on how the approximation error affects the non-convex optimizer; consequently it is not shown that a perturbation optimal for the smoothed operator remains near-optimal (or even produces the desired spectral shift) when the true transfer operator is substituted.
  2. [Numerical examples] Numerical examples section: the ENSO and low-dimensional examples report that the discovered perturbations are 'post hoc natural,' yet no quantitative metric (e.g., table of eigenvalue displacements or correlation-decay rates) compares the effect of the optimal vector field under the kernel-smoothed operator versus the true underlying dynamics or a finer-resolution reference operator.
  3. [Optimal-response vector fields] Definition of optimal-response vector fields: the construction appears to rely on the same kernel-smoothed operator used for the optimization; without an accompanying error analysis or sensitivity test, it is unclear whether these vector fields correctly represent the response of the original nonlinear system under arbitrary observations.
minor comments (2)
  1. Notation for the kernel bandwidth/smoothing parameter is introduced without an explicit symbol in the abstract and early sections; a consistent symbol and discussion of its selection would improve readability.
  2. The manuscript cites Antown et al. (2018) for the core optimization theory; a brief self-contained recap of the relevant theorem (or at least the precise statement used) would help readers who have not consulted the reference.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough and constructive report. We address each major comment below and outline the revisions we will make.

read point-by-point responses

  1. Referee: [Formulation of the optimization problem] the kernel-smoothed operator is used to define the optimization problem, but no a-priori error bound or Lipschitz-type control is given on how the approximation error affects the non-convex optimizer; consequently it is not shown that a perturbation optimal for the smoothed operator remains near-optimal (or even produces the desired spectral shift) when the true transfer operator is substituted.

    Authors: We agree that rigorous a-priori bounds on the propagation of kernel approximation error through the non-convex spectral optimization would be desirable. Deriving such bounds appears technically difficult because the problem is non-convex and the underlying operators are infinite-dimensional. In the revised manuscript we will add a dedicated discussion of this limitation together with numerical sensitivity experiments that vary kernel bandwidth, sample size, and regularization; these tests will quantify how much the computed optimal perturbations and resulting eigenvalue shifts change under controlled perturbations of the approximated operator. revision: partial

  2. Referee: [Numerical examples] the ENSO and low-dimensional examples report that the discovered perturbations are 'post hoc natural,' yet no quantitative metric (e.g., table of eigenvalue displacements or correlation-decay rates) compares the effect of the optimal vector field under the kernel-smoothed operator versus the true underlying dynamics or a finer-resolution reference operator.

    Authors: The observation is correct. For the low-dimensional periodic and chaotic examples we will insert tables that directly compare the eigenvalue displacements and correlation-decay rates obtained from the kernel-smoothed operator against the exact transfer operator (or a high-resolution reference). For the high-dimensional ENSO example, an exact operator is unavailable; we will instead report results obtained from several independent long trajectories and from cross-validation across different kernel parameters to demonstrate consistency of the discovered perturbations. revision: yes

  3. Referee: [Optimal-response vector fields] the construction appears to rely on the same kernel-smoothed operator used for the optimization; without an accompanying error analysis or sensitivity test, it is unclear whether these vector fields correctly represent the response of the original nonlinear system under arbitrary observations.

    Authors: We will revise the section on optimal-response vector fields to include explicit sensitivity tests with respect to kernel bandwidth and data subsampling. We will also add a clarifying paragraph stating that the vector fields are constructed from the data-driven approximation and therefore inherit its limitations; the tests will illustrate the degree of stability of the visualized fields under these variations. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation combines external theory with data-driven approximations

full rationale

The paper constructs kernel-smoothed transfer/Koopman operator approximations from trajectory data and invokes the optimization framework of the external Antown et al. (2018) reference to set up a tractable problem for optimal perturbations. No step reduces a claimed result to a fitted quantity or self-defined input by construction, no load-bearing self-citation chain exists, and the derivation remains self-contained against the stated external benchmark and data-driven inputs. The transfer of optimality from the smoothed operator to the true system is an assumption separate from any definitional circularity.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The approach rests on the applicability of the 2018 theory to kernel approximations and on the choice of kernel and smoothing parameters, which are not specified as fixed or derived.

free parameters (1)
  • kernel bandwidth / smoothing parameter
    Kernel smoothing requires a bandwidth choice that affects the operator approximation; not stated as derived from first principles.
axioms (1)
  • domain assumption The theory developed in Antown et al. (2018) applies directly to the kernel-smoothed operator approximations.
    The optimization problem is formulated by combining the approximations with that external theory.
invented entities (1)
  • optimal-response vector fields no independent evidence
    purpose: Visualising and physically interpreting the system's response to the optimal perturbation under arbitrary observations.
    New interpretive object introduced in the paper; no independent evidence supplied.

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