Universal Approximation Theorems for Dynamical Systems with Infinite-Time Horizon Guarantees

Abel Sagodi; Il Memming Park

REVIEW 2 major objections 4 minor 188 references

Universal Approximation Theorems for Dynamical Systems with Infinite-Time Horizon Guarantees

T0 review · 2 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read This paper proves that finite-size Neural ODEs can approximate Morse-Smale multistable systems and normally hyperbolic continuous attractors over the infinite time horizon: for any ε, δ > 0, all but a δ-fraction of initial conditions stay w

desk verdict First infinite-horizon multistable approximation results for Neural ODEs, with a solid fixed-point theorem, a clever limit-cycle construction, and a load-bearing gap in the continuous-attractor theorem that needs real work. read the letter →

arxiv 2602.08640 v3 pith:66URBFPP submitted 2026-02-09 math.DS q-bio.NC

Abel Sagodi , Il Memming Park This is my paper · ORCID

classification math.DSq-bio.NC MSC 37C1037C2068T07

keywords NeuralODEsuniversalapproximationMorse-Smalesystemsinfinite-timehorizonmultistabilitylimitcyclescontinuousattractorsε-δcloseness

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

Neural ODEs—differential equations whose vector fields are neural networks—have long been known to approximate arbitrary smooth flows on finite windows, but those guarantees degrade or fail for systems with multiple attractors. This paper tries to remove that barrier for a broad, structurally stable class. It claims that a finite-size Neural ODE can track trajectories forever, with error below ε for every initial condition outside a set of volume δ, whenever the target is Morse-Smale with hyperbolic fixed points or hyperbolic limit cycles. For limit cycles it adds exact period matching, because any period mismatch causes unbounded phase drift. For normally hyperbolic continuous attractors—line attractors, ring attractors—it claims the same ε-δ guarantee by tiling the manifold with a dense grid of discrete attractors. A short corollary converts ε-δ closeness into a bound on time-averaged Lp error, linking the topological guarantee to losses used in training.

What carries the argument

The load-bearing machinery is ε-δ closeness, a metric measuring the volume of initial conditions whose supremum-in-time trajectory error exceeds ε; it induces the topology of convergence in measure on the space of flows and is metrizable via the Ky Fan metric. On top of that, the proofs use four tools: (i) structural stability of Morse-Smale flows to confine basin mismatch to a thin separatrix layer of measure below δ; (ii) a transient/asymptotic split bound, Kν/λ, using exponential contraction near hyperbolic attractors; (iii) localized scalar multiplication via bump functions to correct each limit cycle's period exactly, with diagonal dominance of the period Jacobian ensuring that correcti

What would settle it

For a normally hyperbolic disk of fixed points, compute the infimal C1 distance to any perturbation whose hyperbolic fixed points form an ε/3-net of the disk. If this infimum fails to vanish as ε→0, the tiling claim in the continuous-attractor theorem is false; if it vanishes, the proof's Step 1 has a constructive basis.

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Extended reading notes

Core claim

The central claim is a universal approximation theorem for infinite-horizon dynamics: for any target in the class F_FP (Morse-Smale systems whose ω-limit sets are hyperbolic fixed points), and any ε, δ > 0, there exists a finite-size Neural ODE whose flow is ε-δ close to the target on [0, ∞). The same is claimed for F_LC—Morse-Smale systems with hyperbolic limit cycles—provided the hypothesis class supports bump functions so that each cycle's period can be corrected exactly. For F_CA, systems with normally hyperbolic continuous attractors, the claim is achieved by discretizing the attractor: a C1-close proxy with a finite skeleton of hyperbolic fixed points or limit cycles forming an ε/3- or

Load-bearing premise

The continuous-attractor theorems rest on an unproven existence assertion: a C1-arbitrarily-close proxy must exist whose hyperbolic fixed points or limit cycles form a prescribed dense skeleton with spacing ε/3 or ε/8 of the original normally hyperbolic manifold; generic hyperbolicity gives isolated hyperbolic sets, not a skeleton of prescribed spacing, and the paper supplies no construction.

Editorial extensions

If this is right

For any Morse-Smale multistable system with hyperbolic fixed points, a finite Neural ODE can reproduce all trajectories on [0, ∞) within ε except for initial conditions in a δ-measure set; this extends the reach of neural dynamical models beyond globally stable, fading-memory systems.
For hyperbolic limit cycles, exact period matching eliminates unbounded phase drift: trajectories remain within a bounded distance of the true cycle forever, not just for a few periods.
For normally hyperbolic continuous attractors, trajectories stay ε-close forever even though the approximating system has only discrete attractors; the cost is a bounded discretization error controlled by the tiling resolution.
ε-δ closeness over infinite time implies time-averaged Lp error and mean-squared error are bounded by ε^p + δ·D^p, so the topological guarantee transfers to the loss functions used in practical training.
Because the approximation is direct in the same state space rather than an embedding, the attractor geometry and basin structure are preserved, making learned models mechanistically interpretable.

Reading between the lines

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

Extension: The ε-δ metric deliberately sacrifices all information inside the δ-layer—saddle-region timing, exact separatrix geometry, and homoclinic structure. Two networks with identical ε-δ certificates can therefore disagree arbitrarily about trajectories near separatrices, which is precisely the regime where decision-making circuits operate; the paper flags the layer but not this consequence.
Extension: The continuous-attractor proof is only as strong as its Step 1: generic hyperbolicity is invoked where a prescribed dense skeleton is needed, and no construction is supplied. A computational check—computing the minimal C1 distance from a flat normally hyperbolic manifold to a system with an ε-net of hyperbolic attractors—would either supply the missing construction or expose a gap.
Extension: Because exact period matching multiplies the vector field by a bump function, it changes flow speed but not the cycle's geometry or the zero set. This suggests a trainable phase-field parameterization: constraining the network family so that the period is an explicit output, rather than a post-hoc correction, would make the infinite-time guarantee more robust to weight perturbations.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper does something genuinely new: it proves infinite-horizon ε-δ approximation guarantees for multistable continuous-time flows in their native state space, where prior infinite-time results were confined to fading-memory or globally stable systems. The fixed-point theorem (Thm 3) is plausible and well-structured — structural stability confines basin errors to a thin layer, and exponential contraction gives uniform trajectory bounds away from separatrices. The limit-cycle theorem (Thm 4) is the most innovative part: the exact period-matching via localized scaling is a real idea, and the diagonal-dominance argument for multiple cycles is coherent under the stated bump-function assumptions. The paper is also honest about the learnability gap and the fragility of period matching, which I respect.

The continuous-attractor theorem (Thm 5) is where it falls apart a bit. Appendix E Step 1 asserts that Kupka-Smale density gives a C1-close proxy whose hyperbolic fixed points or limit cycles form a prescribed ε/3- or ε/8-net of the NHIM, with stable manifolds controlled. Kupka-Smale genericity does not do that. It gives hyperbolicity and transverse intersections; it says nothing about the number, spacing, or alternation of the resulting invariant set. No construction is supplied. This is a load-bearing gap: without the dense hyperbolic skeleton, there is no proxy to which Theorems 3 and 4 can be applied. It is a gap, not a demonstrated falsehood, but it is exactly where a referee should push.

There is also a smaller but real issue in Theorem 3. The proof defines V = X \ (E_basin ∪ S) and claims uniform separation from separatrices, but E_basin is only contained in the η-neighborhood of S, not equal to it. A point can lie close to a separatrix and still be outside E_basin if it stays in the same basin, and then Lemma 11's uniform-separation premise fails. This is likely repairable by defining the excluded set as the full η-neighborhood of S, but as written the proof has a hole.

Theorem 6 is essentially definitional — the Lp bound follows immediately from the metric — and the paper says as much. The ε-δ metric itself is weaker than it sounds, allowing unbounded error on a δ-measure set, but that is an honest reflection of basin instability.

No data, no code, no circularity. The literature coverage is broad and the self-citations are relevant. Should a serious editor send this to peer review? Yes. The fixed-point and limit-cycle results look worthy of careful verification, and the continuous-attractor theorem needs either a real construction or an explicit conjecture. A good referee should focus on Appendix E and the valid-set separation in Theorem 3.

Referee Report

2 major / 4 minor

Summary. The paper introduces an ε-δ approximation criterion for infinite-horizon trajectory closeness: the volume of initial conditions for which the supremum error exceeds ε is less than δ. It claims universal approximation theorems for Neural ODEs for three target classes: (i) Morse-Smale systems with hyperbolic fixed points, (ii) Morse-Smale systems with hyperbolic limit cycles, using exact period matching by localized vector-field scaling, and (iii) normally hyperbolic continuous attractors, handled by a discretization/tiling strategy. It also proves a temporal generalization bound: if two flows are ε-δ close, then the time-averaged L^p error is at most ε^p + δD^p. The main proofs are in Appendices C–E; Appendix F reviews the literature.

Significance. If fully established, these results would be a substantial step beyond finite-time universal approximation and beyond the monostable/fading-memory infinite-time results. The ε-δ metric is well motivated for multistable systems, and Theorem 6 is a clean and useful bridge to training loss. The period-correction lemmas in Appendix D are largely coherent under the stated bump-function assumptions. However, the central claim for continuous attractors (Theorem 5) depends on an unproven Kupka-Smale-based construction, and the fixed-point theorem (Theorem 3) has a rigor gap in the valid-set argument. Both are fixable but currently prevent the paper from being accepted as a complete proof of its advertised claims.

major comments (2)

[Appendix E, Theorem 5 Step 1] The proof asserts that by the Kupka-Smale Theorem there exists a C^1-arbitrarily-close proxy f† whose fixed points/limit cycles are hyperbolic and form an ε/3- or ε/8-net of the continuous attractor M, with alternating attractors and saddles, and with controlled stable manifolds. Kupka-Smale genericity gives hyperbolicity and transversality for generic perturbations, but it gives no control over the number, spacing, or sink/saddle alternation of the periodic orbits, nor a construction of a skeleton at prescribed mesh size. This assertion is load-bearing: without such a proxy, neither Theorem 3 nor Theorem 4 can be applied to f†, and the entire continuous-attractor claim collapses. The gap is a missing construction or transversality-counting argument, not a demonstrated falsehood, but it must be supplied.
[Theorem 3, full proof Step 3 vs. Lemma 11] The full proof of Theorem 3 takes V = X \ (E_basin ∪ S) and claims that this set has uniform separation η' > 0 from the separatrices. Lemma 11, however, requires exactly inf_{x∈V} dist(x, ∂BoA(A)) ≥ η > 0. E_basin is the set of initial conditions whose basin assignment differs between f and f̂; it is contained in a tubular neighborhood of the separatrices but does not, in general, contain a full neighborhood of S. Thus V may contain points arbitrarily close to the separatrix but not in E_basin, and Lemma 11 cannot be applied as written. The proof can be repaired by defining V as the complement of an η'-neighborhood of S and using vol(N_η'(S)) → 0, but the current text needs to be corrected.

minor comments (4)

[Throughout] There are numerous typographical issues: 'T o' for 'To', 'fail to to a true', inconsistent spacing, and duplicated words. These should be corrected.
[Appendix D, Lemma 18] The proof of Lemma 18 states C^0 closeness of the approximating bump functions, but the hypothesis class has the C^1 UAP. The argument should use C^1 approximation to control the Jacobian off-diagonal bounds, not C^0 alone.
[Remark after Theorem 3] The remark claims vol(E_basin) ≤ C_S ||f - f̂||_C1 without proof. This linear scaling is not used for the main theorem (Lemma 12 already suffices via continuity of measure), so it should be labeled as a heuristic or moved to limitations.
[Appendix E, Case 2 Step 4] The tiling-error bound sup_t dist(γ_θ(t), γ̂_i(t)) ≤ ε/8 + L_H η_base is not justified solely by the ε/8-net property; for two same-period periodic orbits the same-time distance depends on phase alignment. The decomposition should either prove phase alignment or replace this step with a phase-aware estimate.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction found: Theorem 6 is a direct implication of the metric, self-citations are contextual, and the main proofs rely on external theorems. The continuous-attractor discretization assumption in Appendix E is an unproven assertion (a proof gap), not a circular step.

full rationale

I walked the main derivation chain (Theorems 3-6 and Appendices C-E). No load-bearing step reduces to the paper's own inputs by construction. Theorem 3's epsilon-delta guarantee is obtained by combining external structural stability (Palis-Smale), stable-manifold measure-zero separatrix control, and the C1 UAP; the target system is not used to define the approximant. Theorem 4's exact period matching is a genuine construction (bump-function scaling / adjoint first variation) with an external invertibility argument. Theorem 6 is definitional: once Def. 9 fixes the bad set to have volume < delta*vol(X), the Lp bound follows by partitioning X into good/bad sets; this is an implication, not a re-derivation of the theorem's conclusion from a fitted parameter. The self-citations ([131], [142], [143]) are contextual, supporting background remarks on continuous attractors and D-type error, not the epsilon-delta proofs. The most serious weakness is Appendix E Step 1 (and the Case 2 analogue): the paper asserts, from Kupka-Smale genericity, the existence of a C1-arbitrarily-close proxy whose hyperbolic fixed points/limit cycles form an epsilon/3- or epsilon/8-net with controlled basins. That assertion is not supplied and is stronger than the quoted Kupka-Smale theorem. This is a correctness gap that could sink Theorem 5, but it is not circularity: the proxy is assumed, not constructed from the conclusion, and no equation of the paper equates the theorem's input with its output. The manuscript's own limitations sections ('existence vs. learnability', 'fragility of period matching') also weaken practical scope but do not indicate circular reasoning.

Assumptions & free parameters 1 free parameters · 8 assumptions · 0 invented entities

The proofs depend on standard dynamical-systems theorems plus two paper-specific strengthening assertions that are not fully proved: the linear basin-error scaling constant C_S and the existence of a C1-close proxy with a prescribed ε-net of hyperbolic skeleton. No data are fitted; constants such as λ, K, L_T are target-system properties, not free parameters.

free parameters (1)

C_S (basin-error scaling constant) = not estimated
Section 4.2 Remark asserts vol(E_basin) ≤ C_S·||f−f̂||_C1 without proof and uses it to choose η_vol = δ/C_S; no bound or construction is provided.

assumptions (8)

standard math Morse-Smale systems are structurally stable (Palis-Smale).
Invoked in Theorem 3/4 Step 1 to convert C1 closeness into topological equivalence with a homeomorphism close to identity.
standard math Stable manifolds of saddles have measure zero and basin boundaries are finite unions of such manifolds.
Used in Lemma 12 to conclude vol(E_basin) → 0 as the perturbation shrinks.
standard math Feedforward networks with smooth activations have the C1 universal approximation property.
Definition 1 and Theorem 8 (Hornik); the whole hypothesis class depends on this.
standard math Fenichel persistence for normally hyperbolic invariant manifolds.
Theorem 5 uses persistence of M under C1 perturbations with a Hausdorff-distance bound.
ad hoc to paper Kupka-Smale density implies existence of a C1-close proxy whose hyperbolic skeleton is an ε/3- or ε/8-net of the attractor.
Appendix E Step 1: this is the paper's own unproved strengthening of Kupka-Smale; hyperbolicity genericity does not by itself give a prescribed dense skeleton.
ad hoc to paper Basin error volume scales linearly with the C1 perturbation: vol(E_basin) ≤ C_S·||f−f̂||_C1.
Section 4.2 Remark Basin error scaling; stated without proof and used for the main error budget.
domain assumption The hypothesis class supports bump functions and scalar multiplication for limit-cycle corrections.
Theorems 4 and 5 require localized period correction; not true of every neural ODE architecture but achievable with compact-support activations.
domain assumption Continuous attractors in F_CA with oscillatory structure are isochronous.
Definition 13(iv) and the following Remark; without isochrony, the tiling strategy fails due to compounding phase drift.

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Pith. "Pith review of Universal Approximation Theorems for Dynamical Systems with Infinite-Time Horizon Guarantees." pith.science (2026). https://pith.science/paper/66URBFPP

@misc{pith2026260208640,
  author       = {Pith},
  title        = {Pith review of: Universal Approximation Theorems for Dynamical Systems with Infinite-Time Horizon Guarantees},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/66URBFPP}},
  note         = {Machine review of arXiv:2602.08640}
}

abstract

Universal approximation theorems establish the expressive capacity of neural network architectures. For dynamical systems, existing results are limited to finite time horizons or systems with a globally stable equilibrium, leaving multistability and limit cycles unaddressed. We prove that Neural ODEs achieve $\varepsilon$-$\delta$ closeness -- trajectories within error $\varepsilon$ except for initial conditions of measure $< \delta$ -- over the \emph{infinite} time horizon $[0,\infty)$ for three target classes: (1) Morse-Smale systems (a structurally stable class) with hyperbolic fixed points, (2) Morse-Smale systems with hyperbolic limit cycles via exact period matching, and (3) systems with normally hyperbolic continuous attractors via discretization. We further establish a temporal generalization bound: $\varepsilon$-$\delta$ closeness implies $L^p$ error $\leq \varepsilon^p + \delta \cdot D^p$ for all $t \geq 0$, bridging topological guarantees to training metrics. These results provide the first universal approximation framework for multistable infinite-horizon dynamics.

Figures

Figures reproduced from arXiv: 2602.08640 by the authors.

**Figure 1.** illustrates the landscape of universal approximation results. Prior infinitetime theories were limited to fading memory systems (a single global attractor). Our results extend to the full class of Morse-Smale systems and normally hyperbolic continuous attractors143 . non-hyperbolic hyperbolic multi-stability (structurally stable) oscillations marginally stable manifolds infinite-time horizon approximation theorems … view at source ↗

**Figure 3.** The 1D bistable system (Example 1). (a) Vector field f(x) = x − x 3 with stable x± = ±1 (filled) and unstable x0 = 0 (open); phase line shows basins with perturbed separatrix (dashed orange). (b) Trajectories from various initial conditions; orange region marks ICs that switch basins under perturbation ˆf = f − ν cos(2x). (c) Time-averaged L 2 error vs. perturbation size ν: empirical error (black) stays below boun… view at source ↗

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Invertibility via Diagonal Dominance:We construct the Jacobian to be strictly diagonally dominant. The condition|J kk|> P i̸=k |Jki|becomes: ˆTk > X i̸=k ξ ˆTk = (N−1)ξ ˆTk =⇒1>(N−1)ξ. By choosing the bump functions such that the leakageξ < 1 N−1 , the matrixJis strictly diago...
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Support Decay:∥ ˆΦi(x)∥< ζfor allx /∈Mi
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[186]

Substituting this into the integral: Jii =− Z ˜Ti 0 ⟨Zi(t), ˜f(˜γi(t)) +∆ i(t)⟩dt

Diagonal T erms (i=j):On the cycle˜γ i, we write ˆΦi = ˜f+∆ i, where ∥∆i∥< ν. Substituting this into the integral: Jii =− Z ˜Ti 0 ⟨Zi(t), ˜f(˜γi(t)) +∆ i(t)⟩dt. Using linearity and the normalization property⟨Z i(t), ˜f(˜γi(t))⟩ ≡1: Jii =− Z ˜Ti 0 1dt | {z } = ˜Ti − Z ˜Ti 0 ⟨Zi...
[187]

By the leakage assumption,∥ ˆΦj∥< ζon˜γi

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uniform strong universality

Diagonal Dominance:For strict diagonal dominance, we require|J ii|>P j̸=i |Jij|. Substituting our bounds: ˜Ti(1−C Zν)>(N−1) ˜TiCZζ. Dividing by ˜Ti and rearranging: 1> CZν+C Z(N−1)ζ=⇒ν+ (N−1)ζ < 1 CZ . This condition holds by hypothesis, ensuringJis invertible. Proposition 22(...
[2020]

URLhttps://arxiv.org/abs/2012.02414

2012 arXiv
[2024]

URLhttps://openreview.net/forum?id=Ln8ogihZ2S

Pith tools

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

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