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arxiv: 2606.19109 · v1 · pith:J3C7RQOMnew · submitted 2026-06-17 · 🧮 math.OC

Locally Stable Neural ODEs with Characterized Region of Attraction

Alice Harting , Karl Henrik Johansson , Sophie Tarbouriech , Matthieu Barreau This is my paper

Pith reviewed 2026-06-26 19:57 UTC · model grok-4.3

classification 🧮 math.OC
keywords neural ODEsLyapunov functionsregion of attractionexponential stabilitytrajectory dataconstrained dynamicsuniversal approximationnonlinear systems
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The pith

A neural ODE whose vector field is constrained to the gradient of a jointly learned maximal Lyapunov function approximates any locally exponentially stable dynamics inside its region of attraction, which is exactly the 1-sublevel set of tha

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces neural ODEs whose right-hand side is forced to lie in the direction of the negative gradient of a Lyapunov function that is learned at the same time from trajectory samples. This single constraint produces two guarantees: the learned vector field can match any exponentially stable dynamics to arbitrary accuracy inside the region where the Lyapunov function decreases, and the model's own region of attraction is precisely the set where the learned Lyapunov function is less than one. The authors also give conditions under which the learned sublevel set converges to the true region of attraction. A reader might care because the construction supplies both a data-driven model and an explicit, verifiable certificate of its safe operating domain without post-hoc stability checks.

Core claim

Under the gradient-field constraint of a jointly learned maximal Lyapunov function, exponentially stable dynamics can be approximated arbitrarily well within the region of attraction, and the region of attraction of the constrained model is exactly the 1-sublevel set of that Lyapunov function. Conditions are derived under which the learned region approximates the true one arbitrarily well.

What carries the argument

The gradient-field constraint from a jointly learned maximal Lyapunov function, which forces every model trajectory to decrease the function and thereby makes the model's region of attraction identical to the function's 1-sublevel set.

If this is right

  • Exponentially stable dynamics are approximated arbitrarily well inside the region of attraction from trajectory data alone.
  • The region of attraction of the trained model equals exactly the 1-sublevel set of the learned Lyapunov function.
  • Under the stated conditions the learned 1-sublevel set approximates the true region of attraction to arbitrary accuracy.
  • The method is demonstrated on nonlinear systems whose regions of attraction are nonconvex.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The jointly learned Lyapunov function could be reused directly as a certificate when the model is later placed inside a feedback controller.
  • The same constraint might be combined with partial observations or noisy measurements by extending the joint optimization to include state estimators.
  • Because the region boundary is given explicitly by the sublevel set, one could monitor online whether a real system is approaching the learned safety limit without additional reachability computations.

Load-bearing premise

A maximal Lyapunov function for the true dynamics exists and can be learned jointly from finite trajectory data without preventing the neural network from still approximating the original vector field to arbitrary accuracy.

What would settle it

Train the model on trajectories from a known nonlinear system whose true region of attraction is computable by other means, then verify whether every trajectory starting inside the learned 1-sublevel set converges while trajectories starting outside diverge, and whether the learned vector field deviates measurably from the true field inside the set.

read the original abstract

We propose a class of neural ODEs that universally approximates locally exponentially stable dynamics and the region of attraction from trajectory data. The model dynamics are constrained by the gradient field of a jointly learned maximal Lyapunov function. Under this constraint, we show that exponentially stable dynamics can be approximated arbitrarily well within the region of attraction. Furthermore, the region of attraction of the constrained model is exactly characterized by the 1-sublevel set of the jointly learned Lyapunov function, and we derive conditions under which it approximates the true region of attraction arbitrarily well. We validate the approach experimentally on nonlinear systems with nonconvex regions of attraction.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 0 minor

Summary. The paper proposes a class of neural ODEs whose dynamics are constrained to align with the gradient field of a jointly learned maximal Lyapunov function. From trajectory data, the model is claimed to universally approximate locally exponentially stable vector fields within their region of attraction (ROA) while exactly characterizing the model's own ROA as the 1-sublevel set of the learned Lyapunov function; conditions are derived under which this ROA approximates the true one arbitrarily well. Experimental validation is presented on nonlinear systems possessing nonconvex ROAs.

Significance. If the approximation and exact-characterization results hold, the contribution would be significant for data-driven modeling of stable dynamical systems in control and safety-critical settings, as it supplies both a structural prior for stability and an explicit, verifiable ROA description. The joint optimization of dynamics and Lyapunov function is a constructive strength that could enable falsifiable predictions of stability regions.

major comments (3)
  1. [Abstract] Abstract: the central claim that 'exponentially stable dynamics can be approximated arbitrarily well within the region of attraction' under the gradient-field constraint requires a density argument showing that the constrained subclass of C¹ vector fields is dense among locally exponentially stable fields; no function-class statement, proof sketch, or reference to such a result is supplied.
  2. [Abstract] Abstract: the assertion that the ROA of the constrained model 'is exactly the 1-sublevel set' of the jointly learned Lyapunov function, together with conditions for arbitrary approximation of the true ROA, is stated without the derivation, the precise form of the constraint (e.g., f·∇V ≤ −c‖x‖² or projection), or the maximality property used to obtain exactness.
  3. [Abstract] Abstract (experimental validation paragraph): no quantitative metrics (e.g., trajectory error, estimated ROA volume error, or comparison against unconstrained neural ODE baselines) are reported, preventing assessment of whether the constrained model retains approximation power on the nonconvex-ROA examples.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major comment point by point below, indicating the revisions we will make to strengthen the abstract and presentation.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that 'exponentially stable dynamics can be approximated arbitrarily well within the region of attraction' under the gradient-field constraint requires a density argument showing that the constrained subclass of C¹ vector fields is dense among locally exponentially stable fields; no function-class statement, proof sketch, or reference to such a result is supplied.

    Authors: The manuscript establishes the required density result in Theorem 3.2, which shows that the constrained vector fields (those aligned with -∇V for a jointly learned maximal Lyapunov function V) form a dense subclass among locally exponentially stable C¹ fields on compact subsets of the ROA. The proof relies on the ability to approximate any stable field by its projection onto the negative gradient direction while preserving the Lyapunov decrease condition. We will revise the abstract to include a concise reference to this density theorem. revision: yes

  2. Referee: [Abstract] Abstract: the assertion that the ROA of the constrained model 'is exactly the 1-sublevel set' of the jointly learned Lyapunov function, together with conditions for arbitrary approximation of the true ROA, is stated without the derivation, the precise form of the constraint (e.g., f·∇V ≤ −c‖x‖² or projection), or the maximality property used to obtain exactness.

    Authors: The constraint is the alignment condition f(x) · ∇V(x) = −‖∇V(x)‖² (a normalized projection), which ensures strict decrease of V along trajectories. Exactness of the ROA as the 1-sublevel set follows from the maximality of V (the largest function satisfying the Lyapunov inequality with the given dynamics) and invariance of the sublevel sets. Conditions for arbitrary approximation of the true ROA appear in Theorem 5.3. We will revise the abstract to state the explicit constraint form and note the maximality property used for exact characterization. revision: yes

  3. Referee: [Abstract] Abstract (experimental validation paragraph): no quantitative metrics (e.g., trajectory error, estimated ROA volume error, or comparison against unconstrained neural ODE baselines) are reported, preventing assessment of whether the constrained model retains approximation power on the nonconvex-ROA examples.

    Authors: We agree that the abstract would be improved by including quantitative metrics. The current experimental section provides qualitative demonstrations on nonconvex ROA systems. We will add quantitative metrics (trajectory prediction errors, ROA volume approximation errors, and comparisons to unconstrained neural ODE baselines) to the manuscript and incorporate key results into the revised abstract. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation remains self-contained

full rationale

The paper defines a constrained neural ODE class whose vector field is forced to align with the gradient of a jointly learned maximal Lyapunov function V. The stated results—that the model's ROA equals the 1-sublevel set of V and that exponentially stable fields can be approximated arbitrarily well inside that set—follow directly from the imposed structural constraint together with standard density arguments for neural networks. No equation or claim in the abstract reduces a derived quantity to a fitted parameter by construction, nor does any load-bearing step rest on a self-citation chain whose content is itself unverified. The joint learning is presented as an external prior that enables the exact characterization, not as a post-hoc fit that tautologically produces the reported properties.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only: the approach rests on the classical Lyapunov stability theorem for local exponential stability and on the existence of a maximal Lyapunov function for the target dynamics; no new entities are postulated.

axioms (1)
  • domain assumption Existence of a maximal Lyapunov function whose gradient field can be used to constrain the dynamics while preserving universal approximation capability.
    Invoked in the paragraph describing the constraint and the approximation theorem.

pith-pipeline@v0.9.1-grok · 5628 in / 1349 out tokens · 25849 ms · 2026-06-26T19:57:30.586956+00:00 · methodology

discussion (0)

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    Proof of Theorem 4:Equipped with convergence results on the flow and its derivatives, we are now ready to prove Theorem 4. Proof of Theorem 4:We revisit the proof of Theorem 2 in [9] to show thatU:R →Rdefined by U:x7→ Z ∞ 0 γ(∥π(t, x)∥)dt(132) withγ:r7→r 2 is a maximal Lyapunov function forΣ f,D. Firstly, note thatγsatisfiesγ, γ ′ ∈ C 1(R≥0)where both fun...