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REVIEW 5 major objections 3 minor 17 references

Sequentially learning regions of attraction from data

T0 review · 5 major / 3 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Data-driven Lyapunov certificates can be extended iteratively by refining tessellation and data wherever a local certificate failure appears.

desk verdict A useful iterative extension of the SOCP-based Lyapunov certification method, but the abstract overclaims: the refinement loop has no termination proof and can fail whenever the uncertified region leaves the true region of attraction. read the letter →

arxiv 2505.03493 v1 pith:6QQEYDFJ submitted 2025-05-06 eess.SY cs.SY

Oumayma Khattabi , Matteo Tacchi-Bénard , Sorin Olaru This is my paper · ORCID
classification eess.SYcs.SY MSC 93D0593D3093C10
keywords data-drivenstabilityanalysispiecewiseaffineLyapunovfunctionregionofattractioniterativerefinementtessellationsecond-orderconeprogrammingpatchyfunctionsdataset
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 claims that an unknown dynamical system's region of attraction can be certified sequentially from point evaluations of its right-hand side, even when a first data-driven Lyapunov certificate fails in part of the region. The proposed loop re-runs the same optimisation-based piecewise affine (PWA) Lyapunov construction on a smaller, refined domain: it identifies the uncertified polyunion, shrinks the working set around it, densifies the tessellation and dataset, and repeats until the leftover uncertified region lies inside the already-known attracted neighbourhood. Theorem 1 states that, under the nested-inclusion conditions produced by the loop, the outermost level set of the first Lyapunov function is attracted to the equilibrium along the unknown dynamics. A local failure therefore no longer forces a global restart; it becomes the trigger for a targeted local refinement.

What carries the argument

The central mechanism is the nested-inclusion chain of sublevel sets. Each iteration builds a PWA Lyapunov candidate $V_k$ with affine pieces $g_c^T x+b_c$ on triangular cells, computes slack variables $s_{i,c}$ from the second-order cone problem, and forms an uncertified region $C_k$ as the convex hull of $A_k$ and all vertices whose slacks satisfy $s_{i,c}\ge0$. The next iteration's upscaling factor $\beta_{k+1}$ picks $X_{k+1}=\beta_{k+1}C_k$ inside $L^{V_k}_{\alpha_k}$, so the new level set contains the previous failure region and remains inside the previous certificate. This produces a decreasing sequence of invariant sets, and Theorem 1 carries the attractivity of the innermost set back to $L^{V_0}_{\alpha_0}$.

What would settle it

Run the iterative loop on a system whose region of attraction is a known nonconvex set with a cusp or fractal boundary, for example a periodically forced pendulum, using a fixed Lipschitz bound and a coarse initial dataset; if the loop enters its "iteration failed, restart with finer mesh" branch indefinitely and never produces a finite chain satisfying $C_{k-1}\subset L^{V_k}_{\alpha_k}\subset X_k$ for all $k$ up to $C_K\subset A_0$, then the assumed eventual success of refinement under condition (4) would be disproved for that setting.

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Extended reading notes

Core claim

The central claim is an iterative nesting theorem: if each new Lyapunov candidate $V_k$ admits a level $\alpha_k$ with $C_{k-1}\subset L^{V_k}_{\alpha_k}\subset X_k\subset L^{V_{k-1}}_{\alpha_{k-1}}$ for all $k$, and the final uncertified set satisfies $C_K\subset A_0$, then $L^{V_0}_{\alpha_0}$ is attracted to $0$. The loop achieves these inclusions by setting $X_{k+1}=\beta_{k+1}C_k$ with the largest $\beta_{k+1}\geq1$ that keeps $X_{k+1}$ inside $L^{V_k}_{\alpha_k}$, choosing a smaller prior attracted set $A_{k+1}\subset A_k$, refining the triangular tessellation and dataset, and recomputing the PWA Lyapunov function. The numerical section demonstrates the construction on the damped pendulum, where five iterations eliminate the previously uncertified diagonal band; a fully data-driven run that does not use the model to choose the tessellation also converges in one extra step.

Load-bearing premise

The load-bearing premise is that whenever a certificate fails locally, the user can keep refining the mesh and data until the next certificate satisfies the required nesting; the paper instructs restarting with a finer mesh when no scaling factor exists, but it does not prove that this refinement will always succeed.

Editorial extensions

If this is right

  • The method certifies the whole outer level set $L^{V_0}_{\alpha_0}$ as attracted to the equilibrium using only point samples of $f$ and a known Lipschitz bound, not a model.
  • A local failure in the optimisation is reinterpreted as information: it identifies where the data or tessellation under-samples the dynamics, and the next iteration concentrates resources there.
  • Each iteration produces an invariant set strictly inside the previous one, so the certified region of attraction is built by inclusion rather than by one monolithic Lyapunov function.
  • For the damped pendulum, the previously uncertified diagonal band is eliminated after five iterations, and a fully data-driven run that does not use the model to choose the tessellation also converges in one extra step.

Reading between the lines

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

  • The paper leaves discrete and controlled systems as future work, but the proof mechanism suggests the nesting argument extends to discrete-time systems under the analogous one-step Lyapunov inequality.
  • The maximal upscaling factor $\beta_{k+1}$ could double as an adaptive step-size signal: if the best $\beta_{k+1}$ is close to 1, the failure region is near the boundary of the current certificate, suggesting the next refinement should concentrate on that boundary.
  • In the pendulum example the failure is concentrated where large-magnitude data points dominate the optimisation, so one may read the algorithm as an active data-acquisition rule: points near the uncertified vertices matter more than global coverage.
  • A testable prediction is that the number of iterations scales with the geometric complexity of the basin boundary; systems with highly nonconvex or fractal boundaries should need more refinement rounds than systems with polytopic basins.
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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 / 3 minor

Summary. The paper addresses data-driven certification of the region of attraction for an unknown continuous-time dynamical system, assuming only pointwise evaluations of the vector field and a known Lipschitz bound. It builds on the PWA Lyapunov-function construction of [13], in which an SOCP is solved over a tessellation, and proposes an iterative refinement scheme (Algorithm 1): when the optimization leaves some vertices uncertified, the algorithm forms a convex hull of the uncertified region, rescales it, refines the tessellation and dataset, and repeats until the uncertified set is absorbed into the known attracted neighborhood. Theorem 1 states a conditional inclusion-based guarantee for the nested sublevel sets, and the method is demonstrated on a damped pendulum example, including a briefly reported fully data-driven run.

Significance. If the iterative refinement can be guaranteed to make progress, the approach is valuable: it avoids semidefinite programming, produces patchy PWA certificates, and the numerical section includes a fully data-driven run that is independent of the model-seeded run, which is a genuine strength. However, the central contribution as advertised in the abstract is stronger than what is proven. Theorem 1 is a conditional statement that assumes the very nested inclusions that Algorithm 1 is supposed to produce, and no termination or progress guarantee is established for the refinement loop. The manuscript therefore provides a plausible algorithmic framework with a conditional certificate and one numerical demonstration, rather than the 'whenever' theorem promised in the abstract.

major comments (5)
  1. [Section 3.1 / Algorithm 1] The abstract claims that 'whenever those certificates are violated locally, a refinement of the domain and the associated tessellation is produced, thus leading to an improvement'. In Algorithm 1, however, if no β_{k+1} exists with X_{k+1}=β_{k+1}C_k ⊂ L^{V_k}_{α_k}, or if no α_{k+1} exists with C_k ⊂ L^{V_{k+1}}_{α_{k+1}}⊂X_{k+1}, the iteration is declared failed and restarted with a finer tessellation and richer data. No argument, bound, or additional assumption is given that such restarts terminate. In fact, if any C_k intersects the complement of the true region of attraction, no amount of refinement can produce the required Lyapunov certificate, so the algorithm may loop indefinitely without ever satisfying the hypotheses of Theorem 1. This gap between the advertised contribution and the conditional theorem is load-bearing and needs to be addressed, either by proving a progress guarantee under explicit assumptions or by restricting the claims accordingly.
  2. [Section 3.1, Eq. (7) and following paragraph] The statement 'By design of constraint (3e) (see [13, Appendix A] for details), we can certify that ∀x∈X\C, ∇V(x)^T f(x)<0' is essential for Theorem 1 but is not justified in the manuscript. Since C is defined as the convex hull of A and the uncertified vertices, a simplex that has one uncertified vertex is not necessarily contained in C, and the implication from negative slacks at vertices to negativity of ∇V^T f on the whole set X\C is nontrivial. The manuscript should either state the precise imported result from [13, Appendix A] and verify its hypotheses, or provide a self-contained proof; as written, the induction in Theorem 1 rests on an unstated and unchecked implication.
  3. [Algorithm 1] Algorithm 1 is not well-posed in the first loop iteration. The pseudocode tests 'if C_{k−1} ⊈ L^{V_k}_{α_k}' and at k=0 this refers to C_{−1}, which is never defined or initialized. The figures label the first iteration as k=0, and the surrounding text describes a first iteration producing C_0. A base case or initialization must be supplied so that the algorithm can be executed, otherwise the pseudocode as written is ill-posed.
  4. [Theorem 1] The theorem states that the base case is 'C_K = A_K (no non-negative slack variable)', but by equation (7), if no slack is nonnegative then N is empty and C_K = conv(A_K), which equals A_K only when A_K is convex. Assumption 3 only requires A to be a polyunion and does not assume convexity. The proof's statement 'Suppose that K=0. Then, C_0=A_0 i.e. all slack variables in problem (3) are negative' is therefore not valid as written. Either convexity of A must be added to the assumptions, or the theorem and proof must be rephrased to handle C_K = conv(A_K).
  5. [Section 4] The numerical validation is limited in two ways that affect the strength of the paper's central claim. First, the main simulation uses an initial tessellation informed by a known quadratic Lyapunov function for the exact model (Section 3.1), which is a model-based seed rather than a purely data-driven start. Second, the fully data-driven run is only mentioned in a single sentence and in Figure 7, without specifying its initialization, data counts, tessellation, or the number of iterations beyond 'one more iteration'. Since the paper's stated scope is the unknown-dynamics setting, the validation would be substantially stronger if the fully data-driven experiment were described and reported with the same detail as the model-informed run.
minor comments (3)
  1. [Abstract and Section 5] The abstract and conclusion use 'whenever' to describe the refinement guarantee, but Theorem 1 is conditional and Algorithm 1 contains failure branches that are handled heuristically. The wording should be aligned with the actual proven statements.
  2. [Assumption 3 vs. Algorithm 1] Assumption 3 states that 'a fixed dataset D of the system is available', while Algorithm 1 requires generating and refining datasets D_k at each iteration. This apparent inconsistency should be clarified: is the assumption about the initial dataset only, or is the dataset allowed to be augmented?
  3. [Section 2.3, last paragraph] The sentence 'According to [13, Appendix B], a dataset and a tessellation providing enough covering of the set X\A are bound to bring forth a certified Lyapunov function' is informal. Please state the precise theorem from [13, Appendix B] so that the reader can see exactly which sufficient condition is being invoked and in what sense it is not 'sufficient for the algorithm's convergence to optimality'.

Circularity Check

1 steps flagged · score 2.0 of 10

No equational circularity; the main derivation is a conditional induction, with a modest load-bearing self-citation to the authors' prior work [13] and a termination gap that is a correctness concern, not circularity.

  1. self citation load bearing [Section 2.2 and Theorem 1 proof (uses [13, Proposition 6] and [13, Theorem 8])]
    "According to [13, Proposition 6], if the upper bound on ∇V⊤f based on data is negative at all points in X\A where ∇V exists, then for all α∈ R such that L^V_α⊂X, L^V_α is include in the region of attraction."

    The central stability-transfer step — from a data-verified Lyapunov candidate to a certified region of attraction — is imported from [13], a prior paper co-authored by the current second author. Theorem 1 reuses that result at every induction step without reproving it, so the new certification is only as strong as the self-cited certificate. This is a real dependency, but it is not an equation-for-equation reduction of the conclusion to the paper's own inputs: [13] is an externally published result, and the iterative refinement in Algorithm 1 adds independent content, including a fully data-driven simulation. The circularity burden is therefore modest.

full rationale

The derivation chain is largely conditional and data-driven. Theorem 1 is a genuine induction: assuming the nested inclusions C_{k-1} ⊂ L^{V_k}_{α_k} ⊂ X_k ⊂ L^{V_{k-1}}_{α_{k-1}} and C_K ⊂ A_0, each V_k certificate localizes attractivity and [13, Proposition 6]/[13, Theorem 8] supplies the sublevel-set transfer. The iterative refinement is not a fitted re-labeling of the conclusion: the algorithm constructs new Lyapunov functions from fresh data and tessellations, and the stopping criterion uses the known-attracted set A_0 as an independent anchor. The paper's stated failure modes — 'if no such β_{k+1}≥ 1 exists' and 'if no α_{k+1} such that C_k⊂ L^{V_{k+1}}_{α_{k+1}} exists, ... the iteration fails' — are explicitly acknowledged in Algorithm 1 and Section 3.1. No proof is given that restarts with finer tessellations and richer data always succeed, so the abstract's 'whenever' overstates the guarantee; that is a termination/completeness gap, not a circular reduction. The numerical section supports independence by reporting a fully data-driven run that succeeds even without the model-informed tessellation. The only circularity-adjacent point is the heavy reuse of the authors' prior work [13] for the core certificate; since that is an external peer-reviewed result and is not defined in terms of the present paper's output, it raises a small self-citation burden rather than forcing the conclusion by construction.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central guarantee rests on the prior data-based Lyapunov certification of [13], on the domain assumptions of the problem statement, and on the unproven heuristic that refinement always improves the certificate. No free parameter is fitted to reproduce the target region of attraction.

free parameters (4)
  • Lipschitz constant bound M = 2.5
    User-supplied upper bound required by Assumption 1; enters constraint (3e) and learnability condition (4). Chosen for the pendulum example, not fitted to the target region.
  • slack lower bound mu = 5
    Lower bound in constraint (3b); user-chosen numerical parameter controlling the slack minimization.
  • scaling factors beta_k = beta_1=1.98, beta_2=2.26, beta_3=1.8, beta_4=2.32
    Upscaling of the uncertified region C_k to define the next ambient set X_{k+1}; found by search to satisfy inclusion in the level set.
  • initial quadratic Lyapunov matrix Q = [[10,3],[3,4]]
    Used to seed the initial tessellation via PWA approximation of V_Q; hand-chosen and not fitted to the certification target.
assumptions (5)
  • domain assumption f is Lipschitz continuous with known bound M and pointwise evaluations f(x_d) are available.
    Assumption 1; required for the data-based bound in constraint (3e) and for learnability condition (4).
  • domain assumption X contains exactly one equilibrium at 0, which is locally asymptotically stable, and a neighborhood A of 0 is known to be attracted.
    Assumptions 2 and 3; ensures the target is well-posed and A can serve as the terminal set.
  • standard math [13, Proposition 6] is valid: negative data-based upper bound on grad V^T f over X\A implies level sets of V are attracted to 0.
    Imported from prior work by the same group; used in Theorem 1 without re-derivation.
  • domain assumption For a feasible solution of (3), all points outside C = conv(A union {v_i,c : s_i,c >= 0}) are certified to have grad V^T f < 0.
    Stated in Section 3.1 and attributed to [13, Appendix A]; load-bearing for the stopping criterion and Theorem 1.
  • ad hoc to paper Tessellation and data can always be refined to satisfy learnability condition (4) and make the optimization feasible.
    Algorithm 1 restarts with finer tessellation and data on failure; no proof of eventual success is given.

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Cite this review

Pith. "Pith review of Sequentially learning regions of attraction from data." pith.science (2026). https://pith.science/paper/6QQEYDFJ

@misc{pith2026250503493,
  author       = {Pith},
  title        = {Pith review of: Sequentially learning regions of attraction from data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6QQEYDFJ}},
  note         = {Machine review of arXiv:2505.03493}
}
read the original abstract

The paper is dedicated to data-driven analysis of dynamical systems. It deals with certifying the basin of attraction of a stable equilibrium for an unknown dynamical system. It is supposed that point-wise evaluation of the right-hand side of the ordinary differential equation governing the system is available for a set of points in the state space. Technically, a Piecewise Affine Lyapunov function will be constructed iteratively using an optimisation-based technique for the effective validation of the certificates. As a main contribution, whenever those certificates are violated locally, a refinement of the domain and the associated tessellation is produced, thus leading to an improvement in the description of the domain of attraction.

Figures

Figures reproduced from arXiv: 2505.03493 by the authors.

Figure 1
Figure 1. Example case of the simple pendulum According to [13, Appendix B], a dataset and a tessellation providing enough covering of the set X \ A are bound to bring forth a certified Lyapunov function. However, there is no sufficient condition for the density and distribution of the data and vertices to allow the algorithm’s con￾vergence to optimality. Moreover, the complexity of the computations explodes as the number of … view at source ↗
Figure 2
Figure 2. k = 0 (a) Ndk = 349, Nvk = 221 (b) αk = 174, βk = 1.98 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. k = 1 (a) Ndk = 346, Nvk = 238 (b) αk = 146, βk = 2.26 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: k = 2 9 [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: k = 3 (a) Ndk = 370, Nvk = 239 (b) αk = 145, βk = 2.32 [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: k = 4 10 [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: Graphical description of the relationship between the level sets’ boundaries obtained [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]

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Reference graph

Works this paper leans on

17 extracted references · 14 canonical work pages

  1. [13]

    Robustly learning regions of attraction from fixed data,

    M. Tacchi, Y. Lian, and C. Jones, “Robustly learning regions of attraction from fixed data,” IEEE Transactions on Automatic Control , vol. 70, no. 3, pp. 1576–1591, 2025

    work page 2025

  2. [1]

    The general problem of the stability of motion,

    A. M. Lyapunov, “The general problem of the stability of motion,” International Journal of Control, vol. 55, pp. 531–534, Mar 1992

    work page 1992

  3. [2]

    A parametrization of piecewise linear Lyapunov func- tions via linear programming,

    P. Julian, J. Guivant, and A. Desages, “A parametrization of piecewise linear Lyapunov func- tions via linear programming,” International Journal of Control , vol. 72, pp. 702–715, Jan 1999

    work page 1999

  4. [3]

    Estimation of Constraint Admissible Invariant Set with Neural Lyapunov Function

    D. Kim and H. J. Kim, “Estimation of constraint admissible invariant set with neural Lyapunov function,” Sept. 2024. arXiv:2409.19881 [eess]. 11 -1 -0.5 0 0.5 1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Figure 7: Graphical description of the relationship between the level sets’ boundaries obtained iteratively. The progression in terms of set inclusions illust...

    work page Pith review arXiv 2024

  5. [4]

    Actor-critic physics-informed neural Lyapunov control,

    J. Wang and M. Fazlyab, “Actor-critic physics-informed neural Lyapunov control,” IEEE Con- trol Systems Letters, vol. 8, pp. 1751–1756, 2024. arXiv:2403.08448 [cs]

    arXiv 2024

  6. [5]

    An interval approach to compute invariant sets,

    T. Le M´ ezo, L. Jaulin, and B. Zerr, “An interval approach to compute invariant sets,” IEEE Transactions on Automatic Control, vol. 62, pp. 4236–4242, Aug. 2017

    work page 2017

  7. [6]

    Estimating the robust domain of attraction for non-smooth systems using an interval Lyapunov equation,

    A. Goldsztejn and G. Chabert, “Estimating the robust domain of attraction for non-smooth systems using an interval Lyapunov equation,” Automatica, vol. 100, pp. 371–377, Feb. 2019

    work page 2019

  8. [7]

    Learning stable deep dynamics models,

    J. Z. Kolter and G. Manek, “Learning stable deep dynamics models,” Advances in neural information processing systems, vol. 32, 2019

    2019

Show all 17 references
  1. [8]

    Data-driven control with inherent lyapunov stability,

    Y. Min, S. M. Richards, and N. Azizan, “Data-driven control with inherent lyapunov stability,” in 2023 62nd IEEE Conference on Decision and Control (CDC) , pp. 6032–6037, IEEE, 2023

    2023

  2. [9]

    Computing Lyapunov functions using deep neural networks,

    L. Gr¨ une, “Computing Lyapunov functions using deep neural networks,” arXiv preprint arXiv:2005.08965, 2020

    2005 arXiv

  3. [10]

    Computing continuous and piecewise affine Lyapunov functions for nonlinear systems,

    S. F. Hafstein, C. M. Kellett, and H. Li, “Computing continuous and piecewise affine Lyapunov functions for nonlinear systems,” Journal of Computational Dynamics , vol. 2, no. 2, pp. 227– 246, 2016

    2016

  4. [11]

    Data-driven system analysis of nonlinear systems using polynomial approximation,

    T. Martin and F. Allg¨ ower, “Data-driven system analysis of nonlinear systems using polynomial approximation,” IEEE Transactions on Automatic Control , vol. 69, pp. 4261–4274, Jul 2024. Conference Name: IEEE Transactions on Automatic Control

    2024

  5. [12]

    Data-Driven Safe Control of Discrete-Time Non-Linear Systems,

    J. Zheng, J. Miller, and M. Sznaier, “Data-Driven Safe Control of Discrete-Time Non-Linear Systems,” IEEE Control Systems Letters, vol. 8, pp. 1553–1558, 2024. Conference Name: IEEE Control Systems Letters

    2024

  6. [14]

    Stabilization by patchy feedbacks and robustness properties,

    F. Ancona and A. Bressan, “Stabilization by patchy feedbacks and robustness properties,” in Optimal control, stabilization and nonsmooth analysis , pp. 185–199, Springer, 2004

    2004

  7. [15]

    Sur la sph` ere vide.`A la m´ emoire de Georges Vorono¨ ı,

    B. Delaunay, “Sur la sph` ere vide.`A la m´ emoire de Georges Vorono¨ ı,”Proceedings of the Russian Academy of Sciences. Mathematical Series , no. 6, pp. 793–800, 1934

    1934

  8. [16]

    Yalmip : A toolbox for modeling and optimization in matlab,

    J. L¨ ofberg, “Yalmip : A toolbox for modeling and optimization in matlab,” in In Proceedings of the CACSD Conference , (Taipei, Taiwan), 2004

    2004

  9. [17]

    Multi-parametric toolbox 3.0,

    M. Herceg, M. Kvasnica, C. Jones, and M. Morari, “Multi-parametric toolbox 3.0,” in Proc. of the European Control Conference, (Z¨ urich, Switzerland), pp. 502–510, Jul 17–19 2013. http: //control.ee.ethz.ch/~mpt. 13

    2013

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