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When is a System Discoverable from Data? Discovery Requires Chaos

T0 review · 4 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read A dynamical system can be uniquely recovered from its trajectories only when its motion is chaotic; stable systems remain fundamentally ambiguous.

desk verdict Solid continuous-trajectory identifiability results, oversold as finite-observation discovery; worth refereeing with a rewrite of the abstract. read the letter →

arxiv 2511.08860 v2 pith:IC7HFYXD submitted 2025-11-12 math.DS cs.AIcs.LGcs.NAmath.NAnlin.CD

Zakhar Shumaylov , Peter Zaika , Philipp Scholl , Gitta Kutyniok , Lior Horesh , Carola-Bibiane Schönlieb This is my paper
classification math.DScs.AIcs.LGcs.NAmath.NAnlin.CD MSC 37D4537C4534A55
keywords discoverabilitychaostopologicaltransitivityidentifiabilitysetsofuniquenessLorenzsystemfirstintegralsHausdorffdimension
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 tries to establish a foundational condition for data-driven discovery of dynamical systems: uniqueness of the governing equations from trajectory data is possible exactly when the dynamics are chaotic. The authors prove that a system that is chaotic across its whole domain is discoverable from a single trajectory in the space of continuous functions, and that a system chaotic on a strange attractor is analytically discoverable if the attractor's Hausdorff dimension exceeds d−1. They also prove the negative result that any analytic vector field with a nonconstant global analytic first integral—a conserved quantity—cannot be analytically discovered from any trajectory. As a corollary, the Lorenz system, long treated as the standard chaotic benchmark, is analytically discoverable. The upshot is that the systems most amenable to data-driven model discovery are the unstable, chaotic ones, while stable engineering systems are fundamentally underdetermined by trajectory data alone.

What carries the argument

The load-bearing object is the set of uniqueness: a subset A of R^d such that any function in the class V vanishing on A vanishes everywhere. Proposition 3.1 converts discoverability of F from trajectory u into the statement that the trajectory image O = u(U) is a set of uniqueness for V. The analytic case is carried by the dimension criterion: any set with Hausdorff dimension greater than d−1 is an analytic set of uniqueness, since the zero set of a non-zero real-analytic function has Hausdorff dimension at most d−1. Alongside this, topological transitivity, via Birkhoff's theorem, supplies the dense trajectory needed to make a chaotic attractor a set of uniqueness.

What would settle it

A concrete falsifier: find a continuous vector field on a compact metric space that is uniquely identifiable from a single trajectory but is not topologically transitive. The paper's Theorem 4.1 asserts discoverability from one trajectory implies topological transitivity, so such an example would break the equivalence; no such example is given in the paper.

Watch

Extended reading notes

Core claim

The central claim is that discoverability—the existence of a unique vector field F in a function class V consistent with observed trajectories—reduces to a geometric question: does the image of a trajectory form a set of uniqueness for V? In continuous functions, a set is a set of uniqueness iff it is dense, so any topologically transitive system has a dense trajectory and is uniquely identifiable. In real-analytic functions, the relevant criterion is that the trajectory's closure contains a set of Hausdorff dimension greater than d−1, because the zero set of a nonzero analytic function has dimension at most d−1. Chaotic attractors that satisfy this dimension condition—including the Lorenz a

Load-bearing premise

The proofs treat 'data' as the entire continuous trajectory curve u(t) on an open time interval; finite sets of sampled points are never treated, and for analytic functions a finite set has Hausdorff dimension 0 < d−1 and cannot be a set of uniqueness.

Editorial extensions

If this is right

  • Any model fitted to a single dense trajectory of a chaotic system is the true vector field: no other continuous function can reproduce the same trajectory.
  • The Lorenz system is analytically discoverable, so symbolic or neural discovery methods that succeed on it are not merely overfitting—a unique analytic vector field underlies the data.
  • Stable or integrable systems—those with analytic first integrals, such as energy-conserving oscillators—cannot be uniquely identified from any finite number of trajectories, so purely data-driven discovery from them is ill-posed.
  • Finite continuous discoverability forces a decomposition of the state space into finitely many closed invariant cells, each of which is topologically transitive; one trajectory per cell suffices.
  • Known physical priors, such as a conservation law, can restore uniqueness for non-chaotic systems, but only when the law is informative enough to make the candidate vector field isolated.

Reading between the lines

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

  • If the central claim transfers to approximately chaotic or high-dimensional turbulent flows, it would give a theoretical rationale for why weather and climate models learned from data generalize: their trajectories explore enough of the state space to pin down the dynamics.
  • In practice observations are finite samples, not continuous curves; the paper's guarantee is about infinite-resolution data. A testable inference is that finite-sample discoverability should degrade with sample density for chaotic systems, while failing completely for regular ones.
  • The first-integral obstruction suggests that Hamiltonian or conservative systems, abundant in physics, are systematically hidden from purely data-driven discovery; physics-informed constraints are not optional refinements but necessary conditions for uniqueness.
  • The Hausdorff-dimension condition is sufficient, not necessary; there may be analytically discoverable attractors of dimension at most d−1, and searching for such examples would sharpen the boundary of the result.
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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 / 4 minor

Summary. The paper defines discoverability of a vector field F in a function class V as uniqueness of F among all V-members matching a given trajectory (Definition 2.2), and reduces that problem to whether the trajectory image is a set of uniqueness for V (Proposition 3.1). It then proves: in C^0, a single dense trajectory is necessary and sufficient for discoverability (Theorem 4.1); in the real-analytic class, a chaotic attractor that is a set of uniqueness, e.g. by Hausdorff dimension > d-1, gives discoverability from a single trajectory (Theorem 4.13); an analytic first integral obstructs analytic discoverability (Theorem 5.1); and certain conservation laws can restore uniqueness in examples (Corollaries 5.3 and 5.5). The advertised applications are the classical Lorenz system and consequences for data-driven scientific discovery.

Significance. Read as a paper about exact observation of a continuous trajectory on an open time interval, the results are substantial and mostly coherent. The C^0 characterization is clean, the dimension criterion is simple and correct, and the first-integral obstruction is a useful, rigorous negative result. The sheaf-theoretic framework is ambitious and, where completed, provides a promising local picture of non-uniqueness; the worked examples (e.g. Example 4.10) help the reader. The main advertised application, analytic discoverability of the Lorenz system, is conditional on an external dimension result whose hypotheses are not checked in the manuscript. Overall, the paper contains enough correct mathematics to warrant revision, but the central claim as stated in the abstract and introduction is not what is proved.

major comments (4)
  1. [Abstract, §2.1 (Def. 2.2), §3 (Prop. 3.1)] The abstract promises uniqueness from a finite set of observations, but the formal framework observes a differentiable trajectory u:U→R^d on an open interval U, and Proposition 3.1 reduces uniqueness to vanishing on the full image O=u(U). For any finite sample S={x_1,...,x_N}, h(u)=∏_{j=1}^N ‖u−x_j‖² is a non-zero real-analytic function vanishing on S, and the same construction works in C^0. Hence no finite set is a set of uniqueness in either class, regardless of chaos. Theorems 4.1 and 4.13 therefore do not establish the finite-observation claim, and the Section 6 discussion of weather forecasting and digital twins relies on that unproved extrapolation. The authors should either formulate a genuine finite-observation model and prove what can be said, or explicitly reframe the paper's claim as continuous full-trajectory observation.
  2. [§4.2.3, Thm 4.13] The converse in Theorem 4.13 is only sketched. It silently assumes every trajectory converges to an attractor, without stating compact trapping-region or asymptotic-approach hypotheses. The line 'As γ⊆A' appears to be a slip for γ⊆U_A (the basin of attraction), and the extension from one to finitely many trajectories is omitted; one must take the product of the individual non-zero analytic functions and observe that the product is non-zero. Please state the precise assumptions, complete the proof, or downgrade this part to a conjecture. This matters because the theorem is advertised as an 'if and only if' statement.
  3. [§4.2.3, Cor. 4.14] The Lorenz corollary is the paper's headline application, but its proof consists of two citation steps: Moreira et al. [2020] prove dim_H>2 for a class of geometric Lorenz flows, and Tucker [1999,2002] is said to imply the original Lorenz equations belong to that class. Neither the exact statement from [Moreira et al.] nor a verification that the Lorenz system satisfies its hypotheses is provided. Since the advertised result is that the classical Lorenz system is analytically discoverable, the authors should quote the relevant theorem from [Moreira et al.] and confirm that Tucker's computer-assisted proof yields the required geometric model.
  4. [§4.2.1–4.2.2, Lemma 4.5 and Thm 4.9] These results are central to the converse direction and to the non-subanalytic-locus picture, but the proofs are not at the standard of the rest of the paper. In Lemma 4.5, the equivalence 'ϕ_∗O_D coherent iff γ subanalytic' is argued with several unstated finiteness conclusions. In Theorem 4.9, the claim that every x∈Z∩A lies in N is not established: the constructed sequence (t_n) need not diverge or have no convergent subsequence, e.g. for periodic points on the attractor. Please complete these arguments or reduce the statements to what is proved.
minor comments (4)
  1. [Throughout] Several cross-references are inconsistent: 'Theorem 3.1' should be Proposition 3.1, 'Theorem 3.7' should be Proposition 3.7, 'Theorem 4.5' should be Lemma 4.5, and definitions are sometimes cited as theorems (B.5, B.7).
  2. [§2, Def. 2.5 vs Def. 2.2] Definition 2.5 defines a trajectory on [0,+∞], while Definition 2.2 considers an arbitrary open interval U. The relationship should be clarified, especially since flows may not exist for all positive time without additional hypotheses.
  3. [§5, Note 5.2] The note states that Hénon–Heiles and the double pendulum are not chaotic in position-momentum variables and are only chaotic in cross-sections. This is incorrect for the standard Hénon–Heiles system, which has chaotic orbits on the energy hypersurface in phase space. The remark is not needed for Theorem 5.1 and should be corrected or removed.
  4. [Throughout] Typos and small inaccuracies: 'Lorentz' for 'Lorenz', 'Exerciese' for 'Exercise', and the caption of Figure 1 refers to an 'integrable Lorenz model' where the intended contrast is with a non-chaotic system.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation chain is self-contained and built on elementary reductions plus external mathematical results.

full rationale

I walked the derivation chain. Definition 2.2 formalizes discoverability from differentiable trajectories u:U→R^d on an open interval, and Proposition 3.1 reduces uniqueness to the absence of a nonzero G∈V vanishing on O=u(U). Although this reduction is cited to Scholl et al. (overlapping authors), it is a direct equivalence: if two fields fit the trajectory, their difference vanishes on O; conversely, any G vanishing on O gives a competing field F+G. The reduction is elementary and does not smuggle in the paper's conclusions. Proposition 3.5 is proved directly from density of continuous functions, and Lemma 3.10 uses the standard fact that the zero set of a nonzero real-analytic function has Hausdorff dimension ≤ d−1. Theorem 4.13 combines Birkhoff's lemma with the set-of-uniqueness criterion, and Corollary 4.14 relies on external results by Moreira et al. and Tucker, not on this paper's own claims. Theorem 5.1 constructs H=G−c0 from the first integral itself, so non-discoverability follows by direct construction rather than by definitional fiat. The conservation-law recovery results verify explicit Hessian conditions. No fitted parameter is renamed as a prediction, and no load-bearing assertion rests solely on a self-citation. The abstract's phrase 'finite set of observations' is not the formal model—the theorems concern exact continuous trajectories—and the paper itself flags the theoretical-vs-practical gap in Section 6. That is a scope limitation, not a circular reduction.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper is pure mathematical analysis: no parameters are fitted to data. It relies on standard analytic-geometry and dynamical-systems theorems; the only externally quantified input is the Lorenz attractor dimension estimate and the identification of the Lorenz flow with a geometric Lorenz flow.

assumptions (7)
  • standard math Birkhoff's theorem: topological transitivity implies existence of a dense trajectory (hypercyclic point) on complete metric spaces.
    Used throughout Section 4 to convert chaotic transitivity into a dense single trajectory; proved in Appendix A.
  • standard math Identity theorem for real analytic functions: a nonzero analytic function cannot vanish on a nonempty open set.
    Used in Theorem 3.8 and in the proof of Lemma 3.11 to extend local agreement to global agreement.
  • standard math The zero set of a nonzero real analytic function has Hausdorff dimension at most d−1.
    Used in Lemma 3.10 to characterize sets of analytic uniqueness; cites Mityagin 2020.
  • standard math Coherence of the sheaf of real analytic functions on R^d, Cartan's Theorem A, and Oka's coherence theorem.
    Used in Section 4.2.1 and Appendix B to relate trajectory subanalyticity to uniqueness sheaves.
  • standard math Existence and uniqueness of maximal flows for locally Lipschitz vector fields, and forward invariance of trapping regions.
    Used in Section 2 to set up flows, attractors, and basins of attraction.
  • domain assumption Every forward trajectory converges to some attractor and basins of attraction are the relevant non-uniqueness sets.
    Assumed without explicit hypotheses in the converse direction of Theorem 4.13; not true for arbitrary noncompact or escaping flows.
  • domain assumption The Lorenz equations define a geometric Lorenz flow, and geometric Lorenz attractors have Hausdorff dimension strictly greater than 2.
    Used in Corollary 4.14 to conclude the classical Lorenz attractor is analytically discoverable; relies on Tucker (1999, 2002) and Moreira et al. (2020) without verifying all hypotheses in the paper.

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Cite this review

Pith. "Pith review of When is a System Discoverable from Data? Discovery Requires Chaos." pith.science (2026). https://pith.science/paper/IC7HFYXD

@misc{pith2026251108860,
  author       = {Pith},
  title        = {Pith review of: When is a System Discoverable from Data? Discovery Requires Chaos},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IC7HFYXD}},
  note         = {Machine review of arXiv:2511.08860}
}
read the original abstract

The deep learning revolution has spurred a rise in advances of using AI in sciences. Within physical sciences the main focus has been on discovery of dynamical systems from observational data. Yet the reliability of learned surrogates and symbolic models is often undermined by the fundamental problem of non-uniqueness. The resulting models may fit the available data perfectly, but lack genuine predictive power. This raises the question: under what conditions can the systems governing equations be uniquely identified from a finite set of observations? We show, counter-intuitively, that chaos, typically associated with unpredictability, is crucial for ensuring a system is discoverable in the space of continuous or analytic functions. The prevalence of chaotic systems in benchmark datasets may have inadvertently obscured this fundamental limitation. More concretely, we show that systems chaotic on their entire domain are discoverable from a single trajectory within the space of continuous functions, and systems chaotic on a strange attractor are analytically discoverable under a geometric condition on the attractor. As a consequence, we demonstrate for the first time that the classical Lorenz system is analytically discoverable. Moreover, we establish that analytic discoverability is impossible in the presence of first integrals, common in real-world systems. These findings help explain the success of data-driven methods in inherently chaotic domains like weather forecasting, while revealing a significant challenge for engineering applications like digital twins, where stable, predictable behavior is desired. For these non-chaotic systems, we find that while trajectory data alone is insufficient, certain prior physical knowledge can help ensure discoverability. These findings warrant a critical re-evaluation of the fundamental assumptions underpinning purely data-driven discovery.

Figures

Figures reproduced from arXiv: 2511.08860 by the authors.

Figure 1
Figure 1. Model ambiguity in non-chaotic systems versus uniqueness in chaotic systems. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. An illustration of how system dynamics affect model discovery from data. The regular motion of the [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Examples of sets of uniqueness and non-uniqueness for various function spaces. Columns differentiate between function spaces: Linear ([Qiu et al., 2022, Casolo et al., 2025]), Real Analytic (Theorem 3.8), and Continuous (Theorem 3.5). Being a set of uniqueness for a function space implies a function, zero on that set, is zero everywhere. 3.1 The Continuous Case In the continuous case, uniqueness becomes almost trivi… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Proof. Here we assume that we are looking at a continuous flow on a compact metric space X. Discoverable from n trajectories implies that for some {x1, x2, . . . , xn} ⊂ X, we must have by Theorems 3.1 and 3.5 that the union of the closures of their trajectories covers…
Figure 4
Figure 4. Figure 4: Illustration of finite continuous discoverability being equivalent to a decomposition into cells, on each of which [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: Illustration of the main argument behind Theorem 5.1. An analytic first integral foliates the underlying space [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.