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A Data-Driven Framework for Koopman Semigroup Estimation in Stochastic Dynamical Systems

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Pith's one-line read Stochastic Dynamic Mode Decomposition estimates the Koopman semigroup of a stochastic system directly from data through the first-order Galerkin step $\hat{K}_{N,\Delta t,m}=I+\Delta t\,\hat{G}^{-1}\hat{H}$, with claimed convergence as…

desk verdict SDMD is gEDMD's generator estimate plus an explicit Euler step, and the paper's main convergence theorem is for e^{tA_N} rather than the Euler operator actually computed; the experimental validation in the neural-mass example also selects on ground truth. read the letter →

arxiv 2501.13301 v4 pith:KZBTXIOQ submitted 2025-01-23 math.DS

Yuanchao Xu , Kaidi Shao , Isao Ishikawa , Yuka Hashimoto , Nikos Logothetis , Zhongwei Shen This is my paper · ORCID
classification math.DS MSC 37M1060J6047D06
keywords stochasticKoopmanoperatorMarkovsemigroupdynamicmodedecompositionsamplingtimedictionarylearningneuralnetworkbasisselectionspectralanalysisconvergenceguarantees
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 introduces Stochastic Dynamic Mode Decomposition (SDMD), an estimator for the Koopman semigroup of a stochastic differential equation. Instead of recovering the generator and then exponentiating it, SDMD builds the one-step operator $\hat{K}_{N,\Delta t,m}=I+\Delta t\,\hat{G}^{-1}\hat{H}$ from dictionary evaluations, explicitly carrying the sampling time $\Delta t$ in the approximation. The authors claim that this estimate converges in the large-data limit, in the zero-limit of the sampling time, and in the large-dictionary limit, so that its eigenvalues and eigenfunctions approximate those of the true stochastic Koopman semigroup. If correct, the framework gives a numerically stable and computationally cheap route to spectral analysis of noisy dynamics, and its neural-network variant automates basis selection. This matters because stochastic spectra encode decay rates, oscillations, metastable transitions, and hidden slow timescales that deterministic methods miss.

What carries the argument

The load-bearing object is the one-step stochastic Taylor–Galerkin identity $K_{\Delta t}\psi \approx (I+\Delta t A)\psi$, which turns semigroup approximation into a linear least-squares problem whose solution is Eq. (10). The finite-dimensional generator $A_N=P_NAP_N$ together with the empirical Gram matrices $\hat G$ and $\hat H$ carry the spectral content, while the Trotter–Kato theorem is the mechanism that upgrades generator convergence to semigroup convergence as $N\to\infty$. Concentration inequalities for the empirical Gram matrices provide the large-data error bounds.

What would settle it

For an Ornstein–Uhlenbeck process with known spectrum, fix a dictionary and compute the eigenvalues of $\hat K=I+\Delta t\,\hat G^{-1}\hat H$ for decreasing $\Delta t$, comparing them with $e^{\Delta t\lambda_n}$; also compare the $n$-fold iterate $(\hat K)^n$ with $e^{t\hat A_N}$ over fixed time $t=n\Delta t$. If the one-step spectrum does not approach the true semigroup spectrum as $\Delta t\to0$, or the iterates fail to track $e^{t\hat A_N}$ for small $\Delta t$, then the claimed convergence guarantees do not cover the method's actual output.

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Extended reading notes

Core claim

The central claim is that the operator $\hat{K}_{N,\Delta t,m}=I+\Delta t\,\hat{G}^{-1}\hat{H}$, built from empirical Gram matrices $\hat G=(1/m)\Psi_X^*\Psi_X$ and $\hat H=(1/m)\Psi_X^*\Psi_X'$, approximates the stochastic Koopman semigroup $K_{\Delta t}$, not merely its generator. The derivation truncates the stochastic Taylor expansion $K_{\Delta t}\psi_j \approx \psi_j+\Delta t\,A\psi_j$ and solves the resulting least-squares problem by Galerkin projection, so the estimator carries the sampling time explicitly and avoids forming $\exp(\Delta t A)$. The paper claims convergence in three successive limits: large data $m\to\infty$ with explicit concentration bounds, zero sampling time $\Delta t\to0$ toward the generator approximation $A_N=G^{-1}H$, and large dictionary $N\to\infty$ toward the true semigroup via the Trotter–Kato theorem. Consequently the eigenvalues and eigenfunctions of $\hat K$ approximate those of the stochastic Koopman semigroup, and generator eigenvalues are recovered from $\lambda=(\mu-1)/\Delta t$.

Load-bearing premise

The convergence proofs are written for the ideal semigroup built from the approximated generator, but the algorithm actually uses only a one-step linear update; the assertion that iterating that update gives the same semigroup is left unproved.

Editorial extensions

If this is right

  • For small sampling times, SDMD approximates both the semigroup spectrum and, via $\lambda=(\mu-1)/\Delta t$, the generator spectrum; the paper demonstrates this on the Stuart–Landau oscillator, Ornstein–Uhlenbeck process, triple-well potential, and a neural-mass model.
  • The method removes the need to compute matrix exponentials, so spectral estimation costs roughly the assembly and inversion of an $N\times N$ Gram system even when the underlying generator is unbounded.
  • Wrapping the estimator in a neural-network dictionary (SDMD-DL) automates basis selection, and the experiments indicate it can isolate latent slow modulations and transient dynamics that dictionary-learning EDMD misses.
  • Iterating the one-step estimate, $\hat K^n$, yields a discrete-time approximation of the continuous semigroup, so long-horizon predictions can be obtained by repeated application of the same matrix.
  • The claimed guarantees are ordered: for fixed $N$ and $\Delta t$, bounds improve with $m$; then the discretized generator approaches $A_N$ as $\Delta t\to0$; then the Galerkin semigroup approaches the true semigroup as $N\to\infty$.

Reading between the lines

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

  • Editorial inference: because $\hat K=I+\Delta t\,\hat G^{-1}\hat H$ is an explicit Euler step, its iterates inherit a stability constraint; for stiff stochastic systems a practitioner would need $\Delta t$ below a threshold tied to the spectral radius of $A_N$ for reliable long-horizon iteration, while the paper's own experiments analyze the one-step spectrum.
  • Editorial inference: SDMD's accuracy is coupled to how well the drift and diffusion coefficients are known or estimated, and the paper does not quantify how errors in those coefficients propagate into $\hat G$, $\hat H$, and the computed spectrum.
  • Editorial inference: the neural-mass result suggests a testable selection rule—calibrating the dictionary by maximizing one dominant eigenfunction's correlation with a known slow variable may tune the whole spectral approximation; checking this across switching rates and noise levels would turn an observed effect into a criterion.
  • Editorial inference: because the estimator is constructed from i.i.d. initial points and an explicit $\Delta t$, it may transfer more readily than snapshot-pair EDMD to non-stationary or time-inhomogeneous sampling designs, but the paper does not test this.
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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 introduces Stochastic Dynamic Mode Decomposition (SDMD), an estimator of the stochastic Koopman semigroup given by K̂_{N,Δt,m} = I + Δt Ĝ^{-1} Ĥ, where Ĝ and Ĥ are empirical Gram matrices built from dictionary functions and their generator action. The authors claim three convergence guarantees: large-data concentration (Theorem 4.3), zero-sampling-time convergence of the generator approximation (Theorem 4.8), and large-dictionary convergence of the semigroup (Theorem 4.14). A neural-network extension, SDMD-DL, is proposed to learn dictionaries, and experiments on a Stuart-Landau oscillator, an Ornstein-Uhlenbeck process, a triple-well system, and a neural-mass model are reported. The paper's central message is that SDMD provides rigorous, reliable spectral estimates of stochastic Koopman operators while avoiding matrix exponentials.

Significance. If the convergence claims were valid, SDMD would provide a practical and theoretically grounded alternative to gEDMD for stochastic systems, with a computationally cheap one-step update and a neural-network dictionary-learning variant. The paper also makes a valuable effort to connect stochastic Taylor expansion, Galerkin approximation, and Trotter-Kato theory. However, the advertised guarantees do not apply to the object actually computed, and several supporting claims are internally inconsistent. The experimental section is extensive but does not compensate for the theoretical gap. The manuscript, in its current form, does not deliver the rigorous framework it promises.

major comments (5)
  1. [Section 4.3.2 and Remark 4.16] Theorem 4.14 proves strong convergence of the matrix-exponential semigroups e^{tA_N} to e^{tA}, but the SDMD estimator in Eq. (10) is the explicit Euler operator I + Δt Ĝ^{-1}Ĥ, not e^{Δt A_N}. The spectral analyses in Section 6 (for example, Figures 2, 3, and 7, and Table 3) report eigenpairs of this one-step Euler matrix, not of e^{Δt A_N}. Remark 4.16 asserts without proof that iterating (I + Δt A_N)^n converges to the continuous semigroup, appealing vaguely to Trotter-Kato, but the required hypotheses—such as uniform exponential boundedness of the discrete powers and strong consistency of the Euler scheme for the sequence A_N—are neither stated nor verified. This is a load-bearing gap: the rigorous convergence guarantees claimed in the abstract and Section 7 do not apply to the method's actual output.
  2. [Section 5.2, Algorithm 2, and Table 1] The text in Section 5.2 states that SDMD-DL avoids the heavy evaluation of Ψ'_X by using Ψ_Y directly, but Algorithm 2, line 5, explicitly requires computing Ψ'_X(θ), and the updating formula in Table 1 constructs bH(θ) = Ψ_X(θ)^* Ψ'_X(θ). The claimed computational-efficiency advantage over gEDMD-DL is therefore contradicted by the paper's own pseudocode and table. This is a substantive inconsistency because the avoidance of Ψ'_X is presented as a key benefit of the neural-network extension.
  3. [Section 4.2, Theorem 4.8] The zero-sampling-time convergence result is essentially a restatement of the Taylor remainder definition. From Eq. (16), K_{N,Δt} = I + Δt G^{-1}H + o_N(Δt), so A_{N,Δt} - A_N = o_N(Δt)/Δt by Eq. (17). The theorem then follows directly from the defining property of o_N(Δt). No rate, no quantitative bound, and no condition on the uniformity of the remainder are provided. As a result, Theorem 4.8 does not supply a usable convergence guarantee for fixed Δt, which is the regime used in all numerical experiments.
  4. [Theorem 4.14 and Assumption 4.13] The theorem asserts uniform exponential boundedness ∥K^N_t∥, ∥K_t∥ ≤ D e^{ωt} for all N, but the proof only cites [44, Theorem 2.2], which gives individual exponential bounds for each strongly continuous semigroup. The uniformity of the constants D and ω in N is a nontrivial condition required by the Trotter-Kato theorem and is not established. Without it, the uniform-in-N semigroup convergence on compact intervals does not follow from the cited result.
  5. [Section 6.4] The neural-mass experiment acknowledges that SDMD-DL does not converge during training and that the authors select the epoch maximizing the Pearson correlation between the normalized second eigenfunction and the ground-truth latent input. This selection procedure uses information unavailable in practice and is not accompanied by error bars, multiple-seed statistics, or a systematic comparison. The reported superiority of SDMD-DL over EDMD-DL is therefore not a reliable empirical demonstration of the method's robustness.
minor comments (4)
  1. [Eq. (5) and Remark 3.5] Equation (5) states the exact relation μ = e^{tλ}, while Remark 3.5 computes generator eigenvalues as λ = (μ - 1)/Δt, which is only a first-order approximation. The paper should clarify that the reported generator eigenvalues are approximate and discuss the resulting bias, especially in Tables 2 and 3.
  2. [Theorem 4.3] The probability bound contains the quantity ϵ - ∥o_N(Δt)∥_F in the definition of ˜ϵ; for small ϵ the argument of the exponentials may become ill-defined. The assumptions under which the bound is meaningful should be stated.
  3. [Section 4.1, Eq. (16)] The existence of o_N(Δt) = lim_{m→∞} Ψ_X^† o_{m,N}(Δt) is asserted, but no argument is given for why this limit exists independently of the data or why the matrix norm of the limit is o(Δt). This is related to the remainder terms being dropped earlier in Eq. (15) and should be justified.
  4. [General presentation] The manuscript contains numerous typographical and formatting issues, such as the garbled phrase in Section 6.2 ('aware methods (SDMD and gEDMD) for stochastic systems') and inconsistent notation for Δt (sometimes Δ, sometimes ∆t). A careful editorial pass is needed.

Circularity Check

2 steps flagged · score 6.0 of 10

The zero-limit guarantee is a Taylor-remainder tautology, the reported eigen-spectrum is an affine relabeling of gEDMD, and the semigroup theorem covers e^{tA_N} rather than the implemented Euler operator.

  1. self definitional [Section 4.2, Eq. (17), Theorem 4.8 and its proof]
    "Define the matrix AN,Δt as AN,Δt := (KN,Δt − I)/Δt = G−1H + oN(Δt)/Δt, ... From Eq.(17), we know that AN,Δt = AN + oN(Δt)/Δt. Since each element in oN(Δt) is o(Δt) as pointed out in Remark 4.2, we have lim Δt→0 ∥AN,Δt − AN∥ = lim Δt→0 ∥oN(Δt)∥/Δt = 0."

    The matrix KN,Δt was defined in Eq. (16) as I + Δt G^{-1}H + o_N(Δt), where o_N(Δt) comes from the stochastic Taylor expansion K_Δt ψ ≈ ψ + Δt Aψ + o(Δt) used to build the estimator in Eq. (12). Therefore AN,Δt := (KN,Δt − I)/Δt = G^{-1}H + o_N(Δt)/Δt by construction, and the theorem's conclusion is exactly the definition of the o(Δt) remainder. The advertised zero-limit convergence is thus a restatement of Taylor's theorem, not a proof that the implemented estimator K̂ converges to the semigroup e^{ΔtA}.

  2. renaming known result [Section 3.1, Eq. (10) and Remark 3.5; spectral results in Sections 6.2–6.3]
    "bKN,Δt,m := I + Δt bG−1 bH, (10) ... bHv = (μ − 1)/Δt bG v, λ = (μ − 1)/Δt."

    For a fixed dictionary and data, G^{-1}H is the gEDMD generator estimate, as the paper concedes in Remark 3.2. Since K̂ = I + Δt G^{-1}H, every eigenvalue μ of K̂ satisfies (μ−1)/Δt ∈ spec(G^{-1}H). Hence the 'generator eigenvalues' reported by SDMD via Remark 3.5 are exactly the gEDMD eigenvalues under an affine map. The semigroup spectral estimate is therefore a relabeling of the generator estimate from which K̂ was built, not an independent prediction.

full rationale

The large-data concentration bound (Theorem 4.3), the Galerkin generator convergence (Theorem 4.12), and the dictionary-limit framework have independent mathematical content and are not circular: they prove genuine probabilistic and approximation statements, and the experiments compare against analytical spectra of the OU and Stuart-Landau systems. However, one of the three advertised 'rigorous guarantees' — the zero-limit of sampling time — is definitional: K̂ was defined by truncating K_Δt = I + ΔtA + o(Δt), so Theorem 4.8 only divides that Taylor identity by Δt. In addition, Theorem 4.14 establishes convergence for the matrix-exponential semigroup e^{tA_N}, whereas the implemented and experimentally analyzed object is the one-step Euler operator K̂ = I + Δt G^{-1}H; Remark 4.16 asserts without proof that iterating the Euler operator converges to e^{tA}. This is a correctness gap rather than a circular step, but it means the central claim that the computed spectra carry the paper's convergence guarantees is not established. The fixed-dictionary spectral equivalence to gEDMD is disclosed by the authors but still makes the semigroup spectral output an affine renaming of the generator estimate. The paper's self-citations ([21], [59]) are related-work citations and are not load-bearing. Overall, the central 'semigroup convergence' claim is partially circular and partially unsupported, warranting a score of 6.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The method's central formula is a first-order Taylor approximation to e^{ΔtA} composed with the gEDMD generator estimator; it relies on standard semigroup and Galerkin assumptions. No new physical entities are introduced. The main unproved assumption is the uniform exponential boundedness of the approximating semigroups, which Theorem 4.14 announces as a conclusion but does not establish.

free parameters (2)
  • dictionary size N = 20 (OU), 10 (triple-well), 28 (neural mass)
    Chosen by hand per experiment; the convergence claims require N large enough to satisfy Assumptions 4.10 and 4.13, which are not verified.
  • regularization parameter γ = small positive number, exact value not reported
    Used in Ĝ+γI and in the Tikhonov loss; it affects the estimated operator and is not tuned in a reported way.
assumptions (6)
  • domain assumption The Koopman semigroup {K_t}_{t≥0} is a strongly continuous semigroup of bounded linear operators on F (Assumption 2.1).
    Used to define the generator A and to apply semigroup theory including Trotter-Kato in Sections 2 and 4.3.
  • domain assumption The dictionary functions ψ_i and Aψ_i are uniformly bounded on the compact sampling set O (Assumption 4.1).
    Used in the concentration inequalities (Lemma 4.5, Theorem 4.3).
  • domain assumption The Galerkin projections converge to the identity in L^2 and in the graph norm of A (Assumption 4.10).
    Needed for Theorem 4.12; the paper admits this is restrictive for arbitrary dictionaries.
  • domain assumption There exists a core D of A such that A_N f → Af for all f in D (Assumption 4.13).
    Required for the Trotter-Kato argument in Theorem 4.14.
  • domain assumption The first-order stochastic Taylor expansion K_Δt ψ = ψ + Δt Aψ + o(Δt) is valid and the remainder is negligible at the chosen Δt (Eq. 12).
    This is the core approximation of the method; the paper relies on it in Eq. (10) and in Theorem 4.8 via the definition of o_N(Δt).
  • ad hoc to paper The semigroups e^{tA_N} are exponentially bounded uniformly in N with constants D, ω (stated in Theorem 4.14, not proved).
    The theorem announces this as a conclusion, but the proof only cites Pazy's generation theorem for each fixed N, which does not give uniformity over N. This uniformity is assumed without proof and is needed for the conclusion.

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Cite this review

Pith. "Pith review of A Data-Driven Framework for Koopman Semigroup Estimation in Stochastic Dynamical Systems." pith.science (2026). https://pith.science/paper/KZBTXIOQ

@misc{pith2026250113301,
  author       = {Pith},
  title        = {Pith review of: A Data-Driven Framework for Koopman Semigroup Estimation in Stochastic Dynamical Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KZBTXIOQ}},
  note         = {Machine review of arXiv:2501.13301}
}
read the original abstract

We present Stochastic Dynamic Mode Decomposition (SDMD), a novel data-driven framework for approximating the Koopman semigroup in stochastic dynamical systems. Unlike existing methods, SDMD explicitly incorporates sampling time into its approximation, ensuring numerical stability and precision. By directly approximating the Koopman semigroup instead of the generator, SDMD avoids computationally expensive matrix exponential computations, which offers a more efficient and practical pathway for analyzing stochastic dynamics. The framework further integrates neural networks to automate basis selection, which reduces the reliance on manual intervention while maintaining computational efficiency. Rigorous theoretical guarantees, including convergence in the large data limit, zero-limit of sampling time, and large dictionary size, establish the method's reliability. Numerical experiments on canonical stochastic systems validate SDMD's effectiveness in approximating eigenvalues and eigenfunctions of the stochastic Koopman operator.

Figures

Figures reproduced from arXiv: 2501.13301 by the authors.

Figure 1
Figure 1. A flow chart for SDMD method. Remark 3.5. Although in our implementation the inverse of Gˆ + γI in Eq. (10) is computed via Cholesky factorization for numerical stability, the spectral computation can also be formulated as a generalized eigenvalue problem. Specifically, we have Hv b = µ − 1 ∆t G v, λ b = µ − 1 ∆t , 6 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Comparison of eigenvalues of Koopman generator estimated by EDMD and SDMD of the stochastic Stuart [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. OU process: Two tests on computing eigenvalues of Koopman generator [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: OU process: two tests on computing eigenfunctions obtained from SDMD-DL, gEDMD-DL and EDMD-DL. [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]
Figure 8
Figure 8. Figure 8: The eigenvalue λ = 1 corresponds to the system’s steady state. The associated eigenfunction in this case is simply the constant function, which does not provide additional information about the system. It encodes features linked to the long-term behavior of the system.…
Figure 5
Figure 5. Figure 5: OU process: test on computing eigenvalues and eigenfunctions of Koopman generator [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]
Figure 6
Figure 6. Figure 6: 2D Triple Well Potential Landscape 6.4 Recovering Latent Timescales in a Neural Mass Model Our previous examples (Stuart-Landau oscillator, Ornstein-Uhlenbeck process, triple-well potential system) demon￾strated SDMD’s accuracy in recovering known spectral structures. …
Figure 7
Figure 7. Figure 7: Triple-well: Two tests on computing eigenvalues of Koopman operator [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]
Figure 8
Figure 8. Figure 8: Triple-well: two tests on computing eigenfunctions obtained from SDMD-DL and gEDMD-DL. [PITH_FULL_IMAGE:figures/full_fig_p021_8.png]
Figure 9
Figure 9. Figure 9: SDMD recovers latent states and transient dynamics from a stochastic neural-mass model. (A) Phase￾space trajectory of the neural-mass model illustrating two distinct dynamical regimes: a low-activity fixed-point (brown) and a spiral oscillatory state (green), modulated…
Figure 10
Figure 10. Figure 10: Comparison of eigenfunctions estimated by EDMD and SDMD for stochastic Stuart-Landau system with [PITH_FULL_IMAGE:figures/full_fig_p027_10.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

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

  1. ResKoopNet: Learning Koopman Representations for Complex Dynamics with Spectral Residuals

    cs.LG 2025-01 conditional novelty 6.0 of 10

    ResKoopNet learns Koopman eigenpairs by minimizing the spectral residual over neural-network dictionaries, reporting fuller spectra and better latent-state separation than existing Koopman methods.

  2. Reinforcement-Learning-Guided Data-Driven Estimation of Spectral Properties of Stochastic Koopman Semigroups

    math.DS 2025-09 reject novelty 5.0 of 10

    Reinforced SDMD uses RL to select trajectory initial conditions guided by a spectral-consistency reward, with qualitative experiments on three stochastic test systems and standard bandit/DQN/PPO machinery.

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.