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REVIEW 4 major objections 5 minor 31 references

Learning Koopman-based Stability Certificates for Unknown Nonlinear Systems

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read One pipeline learns unknown dynamics, proves stability from data

desk verdict The integration is real and the error-propagation argument is sound, but the advertised formal guarantee for unknown systems rests on oracle quantities (K_f, alpha, and Assumption 4.1) that the pipeline never actually obtains from data. read the letter →

arxiv 2412.02807 v2 pith:CQC6JS7G submitted 2024-12-03 eess.SY cs.LGcs.SYmath.DS

Ruikun Zhou , Yiming Meng , Zhexuan Zeng , Jun Liu This is my paper · ORCID
classification eess.SYcs.LGcs.SYmath.DS MSC 93D3093B3093D2037N35
keywords KoopmangeneratorLyapunovfunctionlearningregionofattractionZubovequationsystemidentificationSMTverificationcontinuous-timenonlinearsystemstrajectorydata
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 proposes a framework that, from trajectory data collected at a relatively low sampling frequency, simultaneously learns the vector field of an unknown continuous-time nonlinear system and learns a Lyapunov function certifying its stability. The key idea is to learn the Koopman generator—the infinitesimal generator of the flow—and use that same learned operator both to reconstruct the dynamics and to solve stability-related PDEs, specifically the Zubov equation, whose solution encodes the domain of attraction. The learned Lyapunov function is then checked with an SMT solver, giving a formal guarantee that a certain sublevel set is a genuine region of attraction for the true, unknown system. If the framework works as claimed, a user with modest data can obtain a certified region of attraction for a black-box system without knowing its equations.

What carries the argument

The load-bearing object is the Koopman generator $L$, defined by $Lh = \nabla h \cdot f$, whose finite-sample approximation is built through a resolvent-type least-squares problem. Because the coordinate functions $x_i$ are included in the dictionary, rows of the same learned generator directly give the identified vector field $\hat{f}$; the same generator and dictionary are reused to solve the linear PDEs $LV = -\eta$ and $LW = -\eta(1-W)$, where the latter is the Zubov equation. Proposition 4.2 is the bridge turning the approximate PDE solution into a certificate: it converts a bound on the residual of the Lyapunov derivative into a bound on the true derivative via the triangle inequality, so the annular condition together with Assumption 4.1 yields a true region of attraction.

What would settle it

Run the framework on a known benchmark system where the true vector field and its Lipschitz constant are known, and deliberately sample too sparsely near the origin so that the identified linearization's sublevel set is not a true region of attraction. If the SMT-verified $\Omega_{c_2}$ still passes verification but simulated trajectories from points inside it do not converge to the origin, the certificate chain has a hole at Assumption 4.1. Alternatively, compare the certified region of attraction with the true domain of attraction on a system with a known maximal Lyapunov function: the ratio of their volumes is a direct quantitative check.

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Extended reading notes

Core claim

The central claim is that a single data-driven pipeline can replace separate system identification and stability analysis: with a shared dictionary of observable functions, the Koopman generator learned from trajectory snapshots yields both an identified vector field and a solution of the Zubov equation. Proposition 4.2 states the precise certificate: if the sample mesh spacing $\delta$, the identification error $\alpha$ on the samples, the Lipschitz constants $K_f$ and $K_{\hat{f}}$ of the true and learned vector fields, and the gradient bound $\nu$ satisfy $((K_f + K_{\hat{f}})\delta + \alpha)\nu < \beta$ on the annular region $c_1 \leq V(x) \leq c_2$, and the learned Lyapunov function $V$ satisfies $\nabla V \cdot \hat{f} \leq -\beta$ there, then the sublevel set $\Omega_{c_2} = \{V \leq c_2\}$ is a region of attraction for the true system, provided Assumption 4.1 holds—that the identified linearization's quadratic sublevel set is already a true region of attraction near the origin. The paper demonstrates on two examples, a reversed Van der Pol oscillator and a two-machine power system, that the certified regions of attraction are less conservative than those from quadratic Lyapunov functions and from a prior neural-network approach, while using far fewer samples.

Load-bearing premise

The guarantee rests on Assumption 4.1: the data-driven model must already be accurate enough near the origin that the quadratic sublevel set of its linearization is a true region of attraction for the real system, and the paper also treats the true Lipschitz constant as known; if either gives way, the certificate only proves convergence to that inner set, not to the origin.

Editorial extensions

If this is right

  • Low-frequency trajectory data suffice: in the polynomial example, 100 sampled initial conditions replace millions of labeled samples used by a neural benchmark.
  • The same learned generator serves two purposes, so identification and stability certification share one computation and one dictionary.
  • Formal SMT verification of the learned function yields a certified region of attraction for the unknown system, not just for the fitted model.
  • Because the Zubov equation is solved rather than only the Lyapunov inequality, the certified region of attraction can approach the true domain of attraction as approximation improves.
  • The framework is modular: if the vector field is identified by another technique, the learned generator can still be used purely for solving the stability PDEs.

Reading between the lines

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

  • If the certified region of attraction is to be trusted for a truly black-box system, the true Lipschitz constant $K_f$ must be estimated or bounded from data, since the paper treats it as known; a practical extension would supply a data-driven Lipschitz bound with its own confidence interval.
  • The gap between the verified condition and the true Lyapunov inequality is controlled by $\beta$, which is reduced by an order of magnitude in the examples; pushing this margin smaller is a quantitative target for data collection and dictionary design.
  • The modular separation suggests a testable comparison: using the learned generator for the Zubov equation versus solving the same PDE directly with a separately identified vector field, to isolate which component gives the larger certified region of attraction.
  • The method's ability to certify outside the sampled region suggests that dictionary choice, not sample count, may be the dominant factor in region-of-attraction quality.
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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

4 major / 5 minor

Summary. The manuscript presents an integrated data-driven framework for simultaneously identifying the vector field of an unknown continuous-time nonlinear system and constructing Lyapunov/Zubov functions, using a learned finite-dimensional approximation of the Koopman generator. The identified generator is used both to approximate the vector field and to solve the Lyapunov and Zubov PDEs in a least-squares sense. The learned Lyapunov function is then verified with SMT solvers, and Proposition 4.2 is stated as the formal bridge: under Assumption 4.1 and bounds on mesh size, model error, Lipschitz constants, and verification margin, a sublevel set of the learned candidate is claimed to be a region of attraction for the true system. Numerical experiments on the reversed Van der Pol oscillator and a two-machine power system report less conservative certified ROAs than quadratic and neural-network baselines.

Significance. The paper addresses a genuine gap: existing Koopman-based Lyapunov constructions either lack formal verification or cannot be integrated with system identification. The proposed reuse of a common dictionary for generator learning and PDE solving is elegant, and the explicit error-propagation statement in Proposition 4.2 is a useful template for data-driven certificates. The authors ship code and use SMT verification, which is a strength. However, the advertised guarantee for unknown systems is conditionally formulated: the certificate requires quantities that the pipeline does not compute from trajectory data (Assumption 4.1, K_f, alpha), and the SMT checks are performed against the learned model rather than the true system. The value of the contribution will be realized once these oracle quantities are replaced by data-driven, verifiable bounds or the claims are reframed accordingly.

major comments (4)
  1. [§IV, Assumption 4.1] Assumption 4.1 is the anchor of Proposition 4.2 and is asserted rather than established. The text states that the assumption 'can typically be satisfied' by collecting data around the origin, and Section VI-A only says 'we verified that Assumption 4.1 indeed holds in this case' without giving a procedure. Because Proposition 4.2's conclusion is convergence to O, not to the origin, failure of Assumption 4.1 would leave only a certificate of convergence to a neighborhood. The paper needs a verifiable data-driven condition (or a rigorous numerical check) that O is a ROA for the true system, or the theorem should be restated with this as an explicit hypothesis and the main claim adjusted.
  2. [§IV, Prop. 4.2; §VI-A, Tables I-II] The formal guarantee for the true system requires quantities that are not available from trajectory data alone. The proof of (10)-(11) uses the true Lipschitz constant K_f and the sampled model error alpha = max_y ||f(y) - f_hat(y)||; Section VI-A states 'We assume that the Lipschitz constant K_f is known,' and Tables I and II set K_f equal to K_f_hat. For a genuinely unknown system these values are oracle information, and underestimating either can violate the inequality ((K_f + K_f_hat)delta + alpha)nu < beta. The paper should provide certified upper bounds on K_f and alpha from the finite trajectory data, or explicitly characterize the theorem as conditional on these data-independent quantities.
  3. [§VI-A, Eq. (9)] The SMT verification in the experiments checks inequality (9) for the learned vector field f_hat, not for the true f. The proof bridges to the true system only through (11), which depends on the global bound on ||f - f_hat|| over the relevant domain. Without a certified bound on that model error, the SMT certificate does not transfer to the unknown system; the experiments therefore demonstrate a certificate for the identified model, not for the true system as claimed in the abstract.
  4. [§III, Theorem 3.1 and Eq. (8)] The paper does not propagate the approximation error from the Koopman generator (Theorem 3.1) and the dictionary truncation to the identified vector field. Theorem 3.1 bounds ||L_{lambda,tau_s} - L_lambda|| on C^1(X), but no corresponding bound on alpha = max ||f - f_hat|| is derived in terms of the number of samples M, the dictionary size N, or the quadrature error. As a result, the main theorem's assumption on alpha is not connected to the actual learning pipeline, and the numerical values of alpha in Tables I-II are computed from the true model.
minor comments (5)
  1. [Assumption 4.1] There is a typo in 'Lyapunov euqation'; it should read 'Lyapunov equation'.
  2. [Theorem 2.5] The statement 'Then D = D if and only if' is circular as written; one of the two D symbols should refer to the domain of attraction explicitly, e.g., 'D = D_A'.
  3. [§VI-A and Figure 1] The caption says 'red dot-dashed circle' for the set X, but the experimental domain X = [-1.2,1.2]^2 is a square; the caption should say 'rectangle' or 'boundary of X' as appropriate.
  4. [§VI-A] The claim that 'beta is reduced by a factor of 1/10' is stated without a comparison baseline; please specify the baseline value and the context so the reader can interpret the improvement.
  5. [§V-C and §VI-A] The first numerical example uses a monomial dictionary selected with prior knowledge that the system is polynomial, which is a form of model information; the paper should clarify that the 'unknown' setting still permits structural priors, otherwise the claim of identifying a fully unknown system is overstated.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the true-system certificate is a genuine perturbation theorem, with unverified oracle inputs as completeness gaps rather than self-referential reductions.

full rationale

The derivation chain is linear and non-circular: trajectory data are used to learn the Koopman generator L, the identified vector field is f_hat = L x via (8), and the Lyapunov/Zubov candidate is obtained by solving (13) or (14) with the same learned generator. The SMT verification in Section VI checks the modified Lyapunov inequality (9) against the learned f_hat. The step from f_hat to the true unknown system is Proposition 4.2, whose proof uses the pointwise error bound (10)-(11) to show that negativity of ∇V·f_hat implies negativity of ∇V·f under the stated hypotheses. This is a genuine implication, not a renaming or a definitional equivalence. The paper's real gaps are completeness gaps, not circularity: it states 'We assume that the Lipschitz constant K_f is known' (Section VI-A), Tables I and II set K_f = K_f_hat and compute α from the true vector field at samples, and Assumption 4.1 is only asserted ('we verified that Assumption 4.1 indeed holds in this case') without giving a data-only verification procedure. Those assumptions are not derivable from the trajectory data as presented, so the advertised formal guarantee for genuinely unknown systems is conditional; but Proposition 4.2 itself is not equivalent to its inputs by construction, and no load-bearing conclusion is obtained merely by citing the authors' prior work. Self-citations to [10], [11], [16], [17], [18], [29], and [30] supply prior learning and verification machinery whose asserted theorems do not contain the present ROA conclusion.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central claim is anchored by the assumed accuracy of the identified generator L, the unverified Assumption 4.1, and known global Lipschitz bounds on the true system. These are not derived from data within the paper; they are conditions the user must supply. The core algorithmic content is a least-squares solve, so the main epistemic debt is the gap between the fitted model and the true system.

free parameters (6)
  • r, scale in eta(x)=r|x|^2 = 0.1
    Chosen for all experiments; eta defines the right-hand side of the Lyapunov and Zubov equations (13)-(14) and therefore shapes the fitted Lyapunov function.
  • monomial dictionary order J=K = 8x8
    Hand-picked for the Van der Pol experiment; the dictionary must be rich enough to approximate the generator and the Lyapunov function, and no convergence study is given.
  • tanh dictionary size = 100 tanh observables plus state variables
    Hand-picked for the power system experiment; random ELM weights are fixed and only theta is learned, so basis size is a hyperparameter.
  • boundary weight lambda_b = 100
    Weight in loss (15) balancing the PDE residual and boundary values; set to 100 in all experiments.
  • Koopman resolvent parameters tau_s, gamma, mu, lambda = 5s, 50 or 10Hz, 2.5 or 3, 1e8
    Sampling horizon, sampling rate, and resolvent regularization constants used to learn the generator; chosen by hand and stated in Tables I and II.
  • verification margin beta = 2.08e-3 for Van der Pol; 8.34e-4 for power system
    Chosen so that beta exceeds nu((K_f+K_hatf)delta+alpha) in Proposition 4.2; the SMT certificate checks inequality (9) with this margin.
assumptions (5)
  • standard math The Koopman family is a C0-semigroup with growth bounds, and the Yosida approximation L_{lambda,tau_s} converges to L as lambda goes to infinity.
    Invoked in Section III to justify learning the generator via the resolvent and the first resolvent identity, following [18].
  • domain assumption The vector field f is locally Lipschitz and trajectories exist globally on a precompact state space X.
    Needed for well-posedness, Lipschitz constants, and the ROA arguments; stated in Section II.
  • ad hoc to paper Assumption 4.1: the identified linearization is accurate enough near the origin that the quadratic sublevel set O is a true ROA for the true system.
    This is the anchor point of the certificate; without it trajectories only converge to O, not to the origin. It is asserted and only informally checked in Section VI-A.
  • domain assumption The Lipschitz constant K_f of the true unknown vector field is known.
    Used in Proposition 4.2 and in the experiments; Section VI-A states 'We assume that the Lipschitz constant K_f is known'. For a truly unknown system this is nontrivial.
  • domain assumption The finite dictionary span captures the generator action and the Lyapunov and Zubov solutions to sufficient accuracy.
    The least-squares formulation (12)-(15) assumes Lh is approximated by Z_N(Ltheta) and that the true Lyapunov function lies near the span of Z_N; no convergence certificate links dictionary size to ROA accuracy.

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Cite this review

Pith. "Pith review of Learning Koopman-based Stability Certificates for Unknown Nonlinear Systems." pith.science (2026). https://pith.science/paper/CQC6JS7G

@misc{pith2026241202807,
  author       = {Pith},
  title        = {Pith review of: Learning Koopman-based Stability Certificates for Unknown Nonlinear Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CQC6JS7G}},
  note         = {Machine review of arXiv:2412.02807}
}
read the original abstract

Koopman operator theory has gained significant attention in recent years for identifying discrete-time nonlinear systems by embedding them into an infinite-dimensional linear vector space. However, providing stability guarantees while learning the continuous-time dynamics, especially under conditions of relatively low observation frequency, remains a challenge within the existing Koopman-based learning frameworks. To address this challenge, we propose an algorithmic framework to simultaneously learn the vector field and Lyapunov functions for unknown nonlinear systems, using a limited amount of data sampled across the state space and along the trajectories at a relatively low sampling frequency. The proposed framework builds upon recently developed high-accuracy Koopman generator learning for capturing transient system transitions and physics-informed neural networks for training Lyapunov functions. We show that the learned Lyapunov functions can be formally verified using a satisfiability modulo theories (SMT) solver and provide less conservative estimates of the region of attraction compared to existing methods.

Figures

Figures reproduced from arXiv: 2412.02807 by the authors.

Figure 2
Figure 2. The learned Lyapunov function and corresponding certified ROA [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 1
Figure 1. The learned Lyapunov function and corresponding certified ROA [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

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