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arxiv: 2606.19669 · v1 · pith:EQSWAHZOnew · submitted 2026-06-18 · 🧮 math.OC · cs.SY· eess.SY

Learning Neural Maximal Lyapunov Functions on mathsf{SO}(n)

Adeel Akhtar , Matthieu Barreau This is my paper

Pith reviewed 2026-06-26 16:45 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY
keywords neural Lyapunov functionsspecial orthogonal grouplogarithmic mapZubov equationregion of attractionLie groupsdynamical systemsstability analysis
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The pith

A neural architecture based on the logarithmic map learns maximal Lyapunov functions for dynamical systems on SO(n).

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

The paper sets out to extend classical Lyapunov analysis, which works in flat Euclidean space, to systems whose states live on the curved manifold SO(n) of rotation matrices. It does so by defining a neural network whose outputs are guaranteed to approximate any continuous function on SO(n) when built around the logarithmic map, then casts the search for the largest possible region of attraction as a Zubov-type partial differential equation on that manifold. The practical step is the derivation of closed-form, computable expressions for the derivative of the log map; these expressions turn the learning task into a two-phase gradient-based algorithm that first finds a rough certificate and then refines it. If the construction holds, control engineers obtain stability certificates for rotational dynamics without having to flatten the geometry or restrict the domain artificially.

Core claim

The central claim is that a neural Lyapunov function constructed from the logarithmic map on SO(n) possesses universal approximation properties, that the maximal region of attraction can be recovered by solving a Zubov-type equation on the group, and that explicit, numerically tractable formulas for the derivative of the log map make gradient training feasible through a two-phase procedure that balances speed and accuracy.

What carries the argument

Neural Lyapunov architecture built on the logarithmic map from SO(n) to its Lie algebra, together with the explicit derivative formulas that enable back-propagation.

If this is right

  • Stability certificates become available for any dynamical system whose state evolves on SO(n) without first embedding the group into a larger Euclidean space.
  • The same architecture and derivative formulas can be reused for any right-invariant vector field on the group once the training data are generated.
  • The two-phase algorithm separates an initial coarse search from a fine-tuning stage, reducing the risk of getting stuck in poor local minima of the loss.
  • Empirical validation on a low-dimensional nonlinear example shows that the learned function indeed certifies a region larger than what a hand-crafted quadratic candidate would provide.

Where Pith is reading between the lines

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

  • The same log-map construction might be adapted to other compact Lie groups once analogous derivative formulas are obtained.
  • Because the architecture is defined intrinsically on the manifold, it could be combined with manifold-aware integrators to produce end-to-end certifiably stable controllers.
  • If the approximation guarantees extend to the learned region boundary, the method supplies a practical way to compute the domain of attraction without solving the Hamilton-Jacobi-Bellman equation directly.

Load-bearing premise

The derived formulas for the derivative of the logarithmic map are both mathematically correct and numerically stable enough to support reliable gradient descent during training.

What would settle it

A concrete low-dimensional SO(n) system for which the two-phase algorithm produces a function whose time derivative along trajectories is positive inside the claimed region of attraction.

read the original abstract

Establishing stability guarantees for dynamical systems on Lie groups is a fundamental challenge, as classical Lyapunov methods developed for Euclidean spaces do not directly transfer to curved geometries. In this paper, we propose a framework for learning maximal Lyapunov functions for systems evolving on the special orthogonal group $\mathsf{SO}(n)$. Theoretically, we introduce a neural Lyapunov architecture based on the logarithmic map with proven approximation capabilities, and we formulate the learning problem via a Zubov-type characterization of the maximal region of attraction. A key technical contribution is the derivation of explicit, numerically tractable formulas for the derivative of the logarithmic map, enabling training through a two-phase algorithm that balances computational efficiency and accuracy. Empirically, we validate the approach on a low-dimensional nonlinear system.

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

1 major / 1 minor

Summary. The paper proposes a neural architecture for learning maximal Lyapunov functions on the Lie group SO(n), constructed via the logarithmic map and equipped with proven approximation properties. The learning problem is cast as a Zubov-type characterization of the maximal region of attraction, and the central technical step is the derivation of explicit, closed-form expressions for the derivative of the matrix logarithm that enable gradient-based optimization through a two-phase training procedure. The framework is illustrated on a low-dimensional nonlinear example.

Significance. If the derivative formulas are verifiably correct and free of hidden singularities, the work supplies a concrete, trainable Lyapunov function class on a non-Euclidean manifold together with a region-of-attraction guarantee; this would constitute a useful bridge between Lie-group geometry and neural Lyapunov methods.

major comments (1)
  1. [Derivative formulas (abstract and the section presenting the explicit expressions)] The load-bearing claim is the explicit derivative of the logarithmic map (stated in the abstract as the key technical contribution enabling the two-phase algorithm). The manuscript must exhibit the full derivation (including any use of the adjoint representation or series expansion of d log) so that it can be checked against the known differential of the matrix logarithm on so(n); without this verification the training procedure rests on an unconfirmed calculation.
minor comments (1)
  1. [Empirical validation] The empirical section reports results only on a low-dimensional system; a second, higher-dimensional example would help demonstrate scalability of the two-phase procedure.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thoughtful review and for highlighting the importance of verifying the derivative formulas. We address the single major comment below and will incorporate the requested changes in the revised manuscript.

read point-by-point responses

  1. Referee: [Derivative formulas (abstract and the section presenting the explicit expressions)] The load-bearing claim is the explicit derivative of the logarithmic map (stated in the abstract as the key technical contribution enabling the two-phase algorithm). The manuscript must exhibit the full derivation (including any use of the adjoint representation or series expansion of d log) so that it can be checked against the known differential of the matrix logarithm on so(n); without this verification the training procedure rests on an unconfirmed calculation.

    Authors: We agree that the full derivation must be exhibited for independent verification. In the revised manuscript we will expand the section on the logarithmic map derivative to include a complete, self-contained derivation that explicitly invokes the adjoint representation of SO(n) and the series expansion of d log, allowing direct comparison with the standard differential-geometric formula on so(n). This addition will be placed immediately before the description of the two-phase training algorithm. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on independent Lie-group calculus and external Zubov theory.

full rationale

The paper introduces a neural Lyapunov architecture on SO(n) using the logarithmic map and derives explicit derivative formulas to enable a two-phase training algorithm, all framed via a Zubov-type maximal region of attraction characterization drawn from prior literature. No step reduces by construction to a fitted input renamed as prediction, a self-definitional loop, or a load-bearing self-citation chain; the derivative formulas are presented as a fresh explicit computation rather than an ansatz imported from the authors' prior work or a renaming of a known empirical pattern. The central claims therefore remain mathematically independent of the paper's own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review limited to abstract; no explicit free parameters, invented entities, or ad-hoc axioms listed beyond standard Lie group and control theory background.

axioms (2)
  • standard math Properties of the logarithmic map on SO(n) allow construction of a neural Lyapunov function with approximation guarantees
    Basis for the neural architecture stated in the abstract.
  • domain assumption Zubov-type characterization correctly identifies the maximal region of attraction for the learning objective
    Used to formulate the learning problem in the abstract.

pith-pipeline@v0.9.1-grok · 5652 in / 1291 out tokens · 43419 ms · 2026-06-26T16:45:59.324963+00:00 · methodology

discussion (0)

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