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Kernel Dynamic Mode Decomposition For Sparse Reconstruction of Closable Koopman Operators

T0 review · 5 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Koopman operators are closable over the Laplacian kernel's function space, giving sparse reconstruction a formal guarantee.

desk verdict The Laplacian-kernel closability claim collapses because the kernel sections are not holomorphic and several key proofs rely on invalid steps; the empirical comparison is suggestive but not enough to carry the paper. read the letter →

arxiv 2505.06806 v1 pith:F24QI4BP submitted 2025-05-11 math.DS stat.ML

Nishant Panda , Himanshu Singh , J. Nathan Kutz This is my paper · ORCID
classification math.DSstat.ML MSC 86A0847N6047N7046E2247B3246E2070G6076F20
keywords DynamicModeDecompositionKoopmanoperatorclosableLaplaciankernelreproducingHilbertspacesparsespatiotemporalreconstructionextendedDMD
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

This paper aims to settle a standing question in kernel-based dynamic mode decomposition: when does the infinite-dimensional Koopman operator, lifted from a nonlinear system onto a reproducing kernel Hilbert space, admit a well-behaved (closable) extension? The authors construct a function space from the Laplacian kernel treated as an $L^2$ density, prove that Koopman operators with affine, contractive symbols are closable and compact on that space, and prove that the Gaussian radial basis function kernel fails the same property. They then run kernel extended DMD with the Laplacian kernel on seven datasets, ranging from Burgers' equation and cylinder flow to Lorenz, Rössler, traffic, and sea-surface temperature, and report that it reconstructs dominant spatial-temporal modes from sparse, irregularly sampled snapshots better than the GRBF kernel. The payoff is a data-driven reconstruction method that carries an operator-theoretic guarantee rather than a purely empirical one.

What carries the argument

The load-bearing object is the RKHS $H_L$ built from the Laplacian measure, with reproducing kernel $K^\sigma(z,w) = \sinh\!\big(\sqrt{\langle z,w\rangle/\sigma^2}\big)/\sqrt{\langle z,w\rangle/\sigma^2}$; the space is assembled by applying the Moore–Aronszajn theorem to an orthonormal basis of holomorphic functions. On this space the Koopman operator is taken with affine holomorphic symbol $\varphi(z)=Az+b$, with $A$ invertible and $0<\|A\|_{\mathrm{Frob}}<1$, so the composition operator is compact; compactness is the mechanism that yields closability, since every compact operator has a closed graph. The argument's second engine is the 0th-snapshot mth-Koopman-mode difference $\Delta^{(0,m)}_{H_L}$, a weighted sum over Koopman modes comparing $K_{F_t}^m\zeta_n(x_0)$ with $\zeta_n(A^m x_0 + b\sum_{i=0}^{m-1} A^i)$; Theorem 7.2 shows this sum converges to zero, which is the paper's definition of faithful reconstruction.

What would settle it

For $D=1$, take $f(z)=\exp(-|z-x|/\sigma)$ and evaluate its Wirtinger derivative $\partial f/\partial\bar z$; it is nonzero almost everywhere, so $f$ is not holomorphic on $\mathbb{C}$. Since $H_L$ was defined to contain only holomorphic functions, this single calculation would disprove the membership asserted in Theorem 4.4 and thereby remove the dictionary from the space where closability is proven.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the RKHS $H_L$ generated by the Laplacian kernel, defined as the space of holomorphic functions square-integrable against $d\mu_L(z) = (2\pi\sigma^2)^{-D}\exp(-\|z\|_2/\sigma)\,dV_D(z)$, supports closable Koopman operators: for every affine holomorphic symbol $\varphi(z) = Az + b$ with invertible $A$ and $0 < \|A\|_{\mathrm{Frob}} < 1$, the composition operator $K_\varphi$ is compact and therefore closable on $H_L$ (Theorem 6.5). The same argument fails for the GRBF kernel, whose RKHS norm forces $\|K_\varphi g_m\|_{H_\sigma} = \infty$ for null sequences $g_m$, so the Koopman operator is not closable there (Theorem 6.7). The paper then defines a 'faithful' Koopman mode decomposition difference $\Delta^{(0,m)}_{H_L}$ that measures the gap between the true dynamical evolution and the RKHS evolution of each Koopman eigenfunction, and proves this difference vanishes identically under the closable affine construction (Theorem 7.2). That vanishing is the theoretical justification offered for using the Laplacian kernel in kernel extended DMD for sparse spatiotemporal reconstruction.

Load-bearing premise

The whole guarantee rests on the Laplacian kernel actually belonging to the space of holomorphic functions the paper builds; the proof only estimates an integral and does not show the kernel is holomorphic in $z$, so if that membership fails the closability result does not cover the dictionary used in practice.

Editorial extensions

If this is right

  • Lap-KeDMD, the Laplacian-kernel version of kernel extended DMD, is guaranteed in the idealized setting to reconstruct dominant spatial-temporal modes from sparse and irregularly timed snapshots, because the reconstruction error defined by the Koopman mode difference vanishes.
  • The GRBF kernel, the standard benchmark, is theoretically disfavored for this task: its RKHS does not support closable Koopman operators, so the same faithfulness guarantee is unavailable.
  • Bounded, compact, closable Koopman operators admit finite-rank approximations, which is the formal justification for approximating the infinite-dimensional operator by a finite matrix in extended DMD.
  • For large state-space dimension $D$, the operator norm gap between the true Koopman evolution and the RKHS evolution can be made smaller than any chosen $\epsilon > (D+1)/(D+1-2\,\mathrm{Tr}(A))$, so the spectral measures of the two operators are close (Theorem 7.1).

Reading between the lines

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

  • The paper's own conclusion notes that the practical side of the faithfulness identity (7.10) is yet to be demonstrated; a numerical study of how fast the partial sums $P_N\Delta^{(0,m)}_{H_L}$ approach zero would supply that missing check.
  • The membership question in Theorem 4.4 is testable directly: since $\exp(-\|z-x\|_2/\sigma)$ is not holomorphic in $z$ on $\mathbb{C}^D$, a Cauchy–Riemann check would settle whether the algorithm's dictionary actually lies inside $H_L$.
  • Because the Laplacian kernel is the $\nu=1/2$ Matern kernel, neighboring low-smoothness Matern kernels may inherit closability, whereas the GRBF failure corresponds to the $\nu\to\infty$ limit.
  • The faithfulness criterion could be reused to audit other kernel choices, such as polynomial or compactly supported kernels, for Koopman reconstruction, not just Laplacian versus GRBF.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 4 minor

Summary. The paper presents a kernel-based extended dynamic mode decomposition (Lap-KeDMD) for sparse spatiotemporal reconstruction, together with an operator-theoretic analysis. The authors construct an RKHS H_L of holomorphic functions square-integrable with respect to a "Laplacian measure" (4.6), claim that the Laplacian kernel sections belong to H_L (Theorem 4.4), establish boundedness and compactness of Koopman operators with affine symbols on H_L (Theorems 5.3 and 6.5), contrast this with an alleged failure of closability for the GRBF kernel (Theorem 6.7), and derive approximation and faithfulness results for Koopman spectral measures and mode decompositions (Theorems 7.1 and 7.2). The paper also reports qualitative comparisons of Lap-KeDMD versus GRBF-KeDMD on seven datasets, including chaotic and PDE examples.

Significance. If correct, the paper would supply a principled operator-theoretic justification for using the Laplacian kernel in KeDMD, addressing a recognized open problem about closability of Koopman operators on RKHSs. The empirical scope (seven datasets, including Burgers equation, flow past a cylinder, Lorenz and Rössler attractors, traffic data, and NOAA SST anomalies) is a genuine strength, and the problem of connecting kernel choice to closable Koopman operators is timely and relevant. However, the central theoretical results contain multiple load-bearing errors: the Laplacian kernel sections are not established to be (and in general are not) holomorphic, so they need not lie in H_L; the closability proof in Theorem 6.5 uses an invalid change of variables and does not verify the norm-convergence condition required for closability; Theorem 7.1 uses a false identity about spectral-measure integrals; and Theorem 7.2 essentially restates the conclusion already built into Definition 2.4. Because these steps are load-bearing for the advertised operator-theoretic guarantee, the paper's central claim is not supported.

major comments (5)
  1. [§4.3, Theorem 4.4] The proof of Theorem 4.4 bounds only the L^2(μ_L) integral of |K^{1,σ}_exp(z,x)|^2 and never establishes that the function is holomorphic. For D=1, K^{1,σ}_exp(z,x)=exp(-|z-x|/σ) is not holomorphic in z on C, and the same failure occurs in higher dimensions, so the Laplacian kernel sections are not members of H_L as defined in (4.8). Consequently, the dictionary used in Algorithm 3.1 is not contained in the Hilbert space on which Theorems 5.3 and 6.5 apply, severing the proposed bridge from the RKHS theory to the Lap-KeDMD algorithm.
  2. [§6.2, Theorem 6.5] The proof's change of variables u=Az+b yields the factor exp(-||A^{-1}(u-b)||^2/σ) dV_D(u)/|det A|, but the next inequality replaces this with exp(-||Au||^2/σ) and then identifies the resulting integral with ||g_m||^2_{H_L}; this identification is false for a general invertible A because dμ_L(u) contains exp(-||u||/σ), not exp(-||Au||/σ). In addition, the proof does not establish closability: Lemma 6.4 requires that g_m→0 in H_L implies K_φ g_m→0 in H_L in norm, whereas (6.3) only shows pointwise convergence through the reproducing kernel. The theorem's conclusion is therefore not derived.
  3. [§7.2, Theorem 7.1, Eq. (7.8)] The identity [∫_D λ dV(λ) g](x) - [∫_D λ dV(λ) g](Ax) = [∫_D λ dV(λ) g](x-Ax) is false: the spectral measure integral is a linear operator applied to the function g, not a function of the evaluation point in such a way that argument differences can be pulled through. The subsequent steps also mix pointwise values with norms and treat |V(ρ(K_Ft))| as a scalar measure despite V being a projection-valued measure. The claimed bound (7.4)–(7.5) is therefore not proven.
  4. [§7.2, Theorem 7.2] The conclusion 𭟋Δ^(0,m)_{H_L} ≡ 0 is already asserted in Definition 2.4, Eq. (2.10), so Theorem 7.2 is a restatement rather than a proof. Moreover, the proof's claim that an infinite series summing to zero has partial sums equal to zero is incorrect; partial sums of a convergent series need not vanish, and the individual terms c_n[K^m_{F_t}ζ_n(x_0)-ζ_n(A^m x_0 + b∑ A^i)] need not be zero. The faithfulness result is thus true by construction, not by the argument given.
  5. [§6.2.1, Theorem 6.7] The proof concludes ∥K_φ g_m∥_{H_σ} = ∞ from an unbounded upper bound on the integral in (6.4). An upper bound that diverges does not imply that the actual norm diverges, so the assertion that K_φ g_m ∉ H_σ is not established. Furthermore, even if some sequence produced an infinite norm, closability failure would require exhibiting a sequence g_m→0 with K_φ g_m converging to a nonzero limit; no such sequence is given. The claimed contrast with the GRBF kernel is therefore unsupported.
minor comments (4)
  1. [Definition 2.5] The definition states X=∅, which contradicts the intended use of a nonempty input space throughout the paper; this should be X≠∅.
  2. [Table 3] The entry for W reads "Field of real, complex and whole numbers respectively"; the phrase "whole numbers" is unclear and should be replaced by the intended set (for example, natural numbers or integers).
  3. [§3.3] The experimental comparison is based on visual inspection of heat maps and error plots; no quantitative summary statistic (such as RMSE, mean absolute error, or correlation) is reported, which makes the claimed superiority of the Laplacian kernel difficult to assess.
  4. [Figure 1] The figure combines unexplained symbols (for example, the script F and the notation Δ^(0,m)_H) and is not self-contained; the caption should define all quantities shown.

Circularity Check

1 steps flagged · score 8.0 of 10

Faithful reconstruction is defined as zero and then re-proved as zero: Theorem 7.2 restates Definition 2.4, so the central 'theoretical justification' for Lap-KeDMD is circular.

  1. self definitional [Definition 2.4 (Section 2.2, Eqs. (2.8)-(2.10)) and Theorem 7.2 (Section 7.2, Eqs. (7.10)-(7.11))]
    "𭟋∆(0,m) H := X n∈N cn [K m Ft ζn(x0)−ζn (A m x0 +b P m−1 i=0 A i )] ... When such a specific operator-theoretic quantification of the Koopman operator is determined subject to its existence over the Hilbert space H, then 𭟋∆(0,m) H = lim N→∞ PN 𭟋∆(0,m) H ≡ 0 (2.10) ... Then the faithful Koopman mode decomposition with respect to the RKHS HL is given as 𭟋∆(0,m) HL = P∞ n=1 cn [K m Ft ζn(x0)−ζn (A m x0 +b P m−1 i=0 A i )] (7.10) ... Lastly, lim N→∞ PN 𭟋∆(0,m) HL = 𭟋∆(0,m) HL ≡ 0. (7.11). Equating both equations with each other yields ... = 0."

    Theorem 7.2's conclusion (7.11) is exactly the assertion (2.10) already built into Definition 2.4, and its formula (7.10) is the same series as (2.8). The proof does not derive the equality of the two Koopman-mode expansions; it 'equates' K_Ft^m ζ_n(x0) with ζ_n(A^m x0 + b Σ A^i), which is precisely the statement that each summand of 𭟋∆ vanishes. Closability/compactness of K_φ gives no reason for K_Ft and K_φ to act identically on ζ_n, nor that the same Koopman modes c_n apply to both expansions. Hence the advertised 'faithful reconstruction' is true by construction rather than by operator theory.

full rationale

The paper contains one genuinely circular, load-bearing step. Definition 2.4 defines the 'faithful' Koopman mode difference by formula (2.8) and then declares, in (2.10), that 𭟋∆(0,m)_H ≡ 0. Theorem 7.2 repeats that same formula as (7.10), repeats the same zero limit as (7.11), and 'proves' it by equating the two Koopman-mode series—an equality that is exactly the difference it is supposed to establish. Closability of K_φ gives no reason for K_Ft^m ζ_n(x0) to equal ζ_n(A^m x0 + bΣ A^i); that equality is assumed when the two expansions are equated. Thus the advertised guarantee that Laplacian-kernel ST reconstruction is 'faithful' is true by construction, not by the operator theory developed in Sections 5-6. Other claimed results are not circular in this sense. Theorem 6.5 is essentially the standard implication compact ⇒ closable, so it is trivial rather than derivational; Theorem 6.7 rests on an external result [29]; and the numerical comparisons are independent benchmarks. Self-citations in the references ([83], [84]) are not load-bearing. I note, without counting them as circularity, that Theorem 4.4 establishes only square-integrability of the Laplacian kernel section against dμ_L and never proves the holomorphy required by (4.8), that the change of variables in the proof of Theorem 6.5 is algebraically wrong, and that Theorem 7.1's choice of ϵ makes the bound vacuous; these are soundness defects, not circularity, but they mean the operator-theoretic bridge to Lap-KeDMD is not independently established.

Assumptions & free parameters 2 free parameters · 6 assumptions · 1 invented entities

The central results rest on several unproven or false premises: the Laplacian kernel's membership in the holomorphic RKHS H_L, the boundedness and compactness of Koopman composition operators on H_L, and the existence of a unitary-type spectral representation for the contraction K_phiA. These are asserted or deferred to the supplementary materials. The empirical superiority claim rests on a fixed kernel bandwidth sigma=1, no code, no seeds, and no quantitative metrics.

free parameters (2)
  • Kernel bandwidth sigma = 1 (used in K1,1_exp and K2,1_exp)
    The reconstruction quality of both Laplacian and GRBF kernels can depend strongly on sigma; the paper fixes sigma=1 for all experiments without tuning or sensitivity analysis.
  • Affine symbol parameters A and b in phi(z)=Az+b = unspecified (only constrained to 0<||A||_Frob<1, A invertible)
    Theorems 6.5, 7.1, and 7.2 depend on this contraction parameter to make K_phi compact, yet no value or relation to the actual dynamics is provided.
assumptions (6)
  • domain assumption Underlying dynamical system has flow F_t and invariant Borel probability measure mu (Assumption 2.1)
    Needed to define the Koopman operator on L2(Omega, mu); this is standard in the DMD literature.
  • domain assumption Observables zeta_n belong to L2(Omega) intersect H_L and are Koopman eigenfunctions in both spaces (Section 7 setup)
    This intersection and eigenfunction assumption is strong and unverified for the empirical datasets; the paper does not show the data observables lie in H_L.
  • ad hoc to paper Koopman operator on H_L is bounded and compact for affine contraction symbols (Theorem 5.3, Note 5.2)
    The proof of compactness is deferred to the Supplementary Materials, and the proof of boundedness in Theorem 6.5 contains an incorrect change-of-variables step.
  • ad hoc to paper The spectral integral representation integral lambda dV(lambda) applies to K_phiA on H_L in Theorem 7.1
    K_phiA is assumed compact and contractive, not unitary, and no spectral measure on the unit circle is established for it; equation (7.7) is asserted without proof.
  • ad hoc to paper The Laplacian kernel K1,sigma_exp(.,x) belongs to H_L (Theorem 4.4)
    The proof only checks L2 integrability; it does not show the function is holomorphic. exp(-||z-x||/sigma) is not holomorphic in z on C^D, so the claim is false.
  • standard math Moore-Aronszajn and Bochner theorems
    Used to construct the RKHS and to represent stationary kernels; standard background results.
invented entities (1)
  • Hilbert space H_L with reproducing kernel K_sigma(z,w) = sinh(sqrt(<z,w>/sigma^2))/sqrt(<z,w>/sigma^2)
    purpose: To serve as the RKHS that makes Koopman operators closable and to justify using the Laplacian kernel in KeDMD
    The paper introduces H_L as the RKHS of the Laplacian kernel, but its reproducing kernel is not the Laplacian kernel; the Laplacian kernel is also not holomorphic, so the identification is unsupported and no external evidence is provided.

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Pith. "Pith review of Kernel Dynamic Mode Decomposition For Sparse Reconstruction of Closable Koopman Operators." pith.science (2026). https://pith.science/paper/F24QI4BP

@misc{pith2026250506806,
  author       = {Pith},
  title        = {Pith review of: Kernel Dynamic Mode Decomposition For Sparse Reconstruction of Closable Koopman Operators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F24QI4BP}},
  note         = {Machine review of arXiv:2505.06806}
}
read the original abstract

Spatial temporal reconstruction of dynamical system is indeed a crucial problem with diverse applications ranging from climate modeling to numerous chaotic and physical processes. These reconstructions are based on the harmonious relationship between the Koopman operators and the choice of dictionary, determined implicitly by a kernel function. This leads to the approximation of the Koopman operators in a reproducing kernel Hilbert space (RKHS) associated with that kernel function. Data-driven analysis of Koopman operators demands that Koopman operators be closable over the underlying RKHS, which still remains an unsettled, unexplored, and critical operator-theoretic challenge. We aim to address this challenge by investigating the embedding of the Laplacian kernel in the measure-theoretic sense, giving rise to a rich enough RKHS to settle the closability of the Koopman operators. We leverage Kernel Extended Dynamic Mode Decomposition with the Laplacian kernel to reconstruct the dominant spatial temporal modes of various diverse dynamical systems. After empirical demonstration, we concrete such results by providing the theoretical justification leveraging the closability of the Koopman operators on the RKHS generated by the Laplacian kernel on the avenues of Koopman mode decomposition and the Koopman spectral measure. Such results were explored from both grounds of operator theory and data-driven science, thus making the Laplacian kernel a robust choice for spatial-temporal reconstruction.

Figures

Figures reproduced from arXiv: 2505.06806 by the authors.

Figure 1
Figure 1. The paradigm of interpret-ability of a dynamical system through Kernel Extended Dynamic Mode Decompositions. We will incorporate the Laplacian kernel to overcome our Experimental Aim 1.1 whose Koopman operator theoretic inference is given in the last couple of equations capturing Koopman mode de￾composition difference. As far as ST reconstruction is concerned, ML-centered architectures [60] (ODE-RNNs), [34,53,78] an… view at source ↗
Figure 2
Figure 2. The lifting perspective of the Koopman operators. 2.1 Koopman Mode Decomposition Recall the Borel measurable set against the Borel measure µ in the observable space F = L 2 (Ω, µ) = L 2 (Ω). These systems, in fact, are general and common in practice due to the phenomena energy preserving and volume preserving with respect to µ over Ω [3, 20, 33, 36, 75]. Furthermore, we can also assume that the dynamical system cons… view at source ↗
Figure 3
Figure 3. Pairwise respective surface and contours plot of kernel functions K1,σ exp and K2,σ exp. Definition 2.8 (Reproducing Kernel Hilbert Space). Let X = ∅ and (H,⟨·, ·⟩H) be the Hilbert function space on X. 1. The space H is called as the reproducing kernel Hilbert space (RKHS) if ∀x ∈ X, the evaluation functional Ex : H → K defined as Ex(f) := f(x), f ∈ H is continuous. 2. A function k : X × X → K is called reproducing … view at source ↗
Figures from the paper (24 more)
Figure 4
Figure 4. Figure 4: An illustration of notion of observables for the Burger Equation’s data (middle) when we chose the dictionary of Laplacian kernel (left) and GRBF kernel (right). We briefly recall the kernel-extended DMD algorithm as follows. Suppose [x1, x2, . . . , xM] ∈ R N×M be M d…
Figure 5
Figure 5. Figure 5: Experiment 1: Data matrix of non-linear Burger’s Equation. 3.2 Experimental evaluations For all seven experiments under the assumption of Ex￾perimental Aim 1.1, we provide the following details: 1. For the experimental simulation, the irregular and sparse format of the…
Figure 6
Figure 6. Figure 6: Spatial-temporal reconstruction of the snapshot #39 in the format of data matrix for non-linear Burger’s Equation. After this, the following figure shows the spatial values of the total counts 256, and the reconstruction for the snapshot #39 by both kernels. (a) Spatia…
Figure 7
Figure 7. Figure 7: Spatial-values comparison of the snapshot #39 for non-linear Burger’s Equation along with EWE plot. 3.2.2 Experiment 2 For this experiment, we provided only 100 snapshots out of the available 151 snapshots for the spatial-temporal reconstruction via kernel-extended DMD…
Figure 8
Figure 8. Figure 8: Spatial-temporal reconstruction of the snapshot #99 in the format of data matrix for periodic fluid flow across cylinder. After this, the following figure shows the spatial values of the considered counts 100 out of the total counts available 89, 351, and the reconstru…
Figure 9
Figure 9. Figure 9: Spatial-values comparison of the snapshot #99 for the fluid flow across cylinder. 3.2.3 Experiment 3 Dataset for this experiment built with an initial condition given as [−1.8760, 1.7868]⊤. This experiment belongs to the natural setting of the phase-portrait reconstruc…
Figure 10
Figure 10. Figure 10: Reconstruction of the snapshot #39 for the chaotic Duffing’s Oscillator. 3.2.4 Experiment 4 For the Seattle I-5 freeway traffic speed data, there are 75 loop detectors that record vehicle speed in a 72 time interval frame during a rush hour period from 6:00 am to 12:0…
Figure 11
Figure 11. Figure 11: Experimental information for the Seattle I-5 Traffic data set. We provide the histogram distribution of speed data for the Seattle I5 freeway collected through loop detector IDs 166th, 186th and 226th followed by their respective graphs [PITH_FULL_IMAGE:figures/full_…
Figure 12
Figure 12. Figure 12: Histogram distribution and plot of speed data on three loop detector ID’s on Seattle I5 freeway. We now present the results of the spatial-temporal reconstruction and their error plots of Seattle I5 traffic data for the sensor ID’s 218ththrough the kernel-extended DMD…
Figure 13
Figure 13. Figure 13: Reconstruction for the Seattle I-5 highway sensor 218th . (a) Spatial values (b) EWE plot [PITH_FULL_IMAGE:figures/full_fig_p017_13.png]
Figure 14
Figure 14. Figure 14: Reconstruction result for the Seattle I-5 freeway traffic sensor ID 218th . 3.2.5 Experiment 5 For the following experiment of the Lorenz 1963 three-dimensional dynamical system, we now provide the first three important characteristic plots of the system [PITH_FULL_I…
Figure 15
Figure 15. Figure 15: Lorenz 1963 Attractor followed by its three two dimensional attractor. In the following collection of figures, we provide the time series information of the Lorenz 1963 data. (a) x dimension (b) y dimension (c) z dimension [PITH_FULL_IMAGE:figures/full_fig_p018_15.png]
Figure 16
Figure 16. Figure 16: Lorenz 1963 time series all across three dimension. We now present the results of the spatial-temporal reconstruction and their error plots of the Lorenz 1963 dynamical system through the kernel-extended DMD by both kernels. The results shown below correspond to the s…
Figure 17
Figure 17. Figure 17: Spatial reconstruction result for the Lorenz 1963 dynamical system. (a) Spatial values (b) EWE plot [PITH_FULL_IMAGE:figures/full_fig_p019_17.png]
Figure 18
Figure 18. Figure 18: Spatial reconstruction result for the Lorenz 1963 dynamical system. 3.2.6 Experiment 6 For the following experiment, we will provide first the initial details of the experiment as we did for the Lorenz 1963 dynamical system. (a) R¨ossler attractor (b) 2D xy attractor …
Figure 19
Figure 19. Figure 19: R¨ossler Attractor followed by its three two dimensional attractor. Now, we provide the time series evolution of the R¨ossler attractor with respect to time [PITH_FULL_IMAGE:figures/full_fig_p019_19.png]
Figure 20
Figure 20. Figure 20: R¨ossler Attractor time series across all three dimensions. Now, we are ready to provide the kernel-extended DMD result yielded by the choice of both kernels. (a) Actual (b) Reconstruction by K 1,1 exp (c) Reconstruction by K 2,1 exp [PITH_FULL_IMAGE:figures/full_fig…
Figure 21
Figure 21. Figure 21: Spatial values reconstruction for R¨ossler Attractor. The following is now the spatial plots and error plots as well. (a) Spatial values (b) EWE plot [PITH_FULL_IMAGE:figures/full_fig_p020_21.png]
Figure 22
Figure 22. Figure 22: Spatial values reconstruction for R¨ossler Attractor followed by error plot. 3.2.7 Experiment 7 The following data are from the US Department of Commerce of the National Oceanic and Atmospheric Administration (NOAA). We are interested in the reconstruction of the mean…
Figure 23
Figure 23. Figure 23: NOAA Global Sea Surface Temperature Anomaly for year 2023. The following sets of results present the reconstructed results corresponding to the 9th snapshot which corresponds to the year 2023. (a) Actual (b) Reconstruction by K 1,1 exp (c) Reconstruction by K 2,1 exp …
Figure 24
Figure 24. Figure 24: Reconstruction of NOAA Global Sea Surface Temperature anomalies across latitude and longitude (36 × 72) for the year 2023 [PITH_FULL_IMAGE:figures/full_fig_p021_24.png]
Figure 25
Figure 25. Figure 25: Reconstruction of values of NOAA Global Sea Surface Temperature anomalies and its comparison for the year 2023. 3.3 Experimental Inference We presented Lap-KeDMD to discover modal information from a given limited spatial-temporal data set and compared the results in c…
Figure 26
Figure 26. Figure 26: Schematic presentation of important aspects of GRBF kernel in relation with SE covariance function and the Gaussian measure. GRBF kernel evaluated along (z, −z) provides the GRBF measure which leads to RKHS Segal-Bargmann-Fock Space with exponential dot product kernel…
Figure 27
Figure 27. Figure 27: An illustration of the spectral measure approximations of the data-driven Koopman operators over L 2 (Ω) due to the closability of the function theoretic Koopman operators over the RKHS HL. Note that, this is under an assumption that the operator-theoretic Koopman ope…

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