arxiv: 2605.09546 · v1 · submitted 2026-05-10 · 📡 eess.SY · cs.SY

Recognition: 2 theorem links

· Lean Theorem

PolarNet: Single-Minima Neural Network for Modeling Lyapunov Functions

Hefu Ye, Jiaxin Cheng, Yicong Zhou, Yuan Zhong

Pith reviewed 2026-05-12 03:51 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords neural networksLyapunov functionsstabilitycontrol systemssingle critical pointneural Lyapunov controlproper functionsuniversality

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The pith

PolarNet is a neural network architecture that structurally guarantees a single critical point for modeling Lyapunov functions.

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

Standard neural networks trained to approximate Lyapunov functions can develop multiple critical points, which disrupts the ability to certify stability in learned control systems. PolarNet is constructed so that its output has exactly one critical point at the origin by design. The authors prove that functions represented by PolarNet are proper, growing without bound as states increase, and are universal in the sense that they can approximate any suitable Lyapunov function. Replacing standard networks with PolarNet in existing training procedures avoids particular failure modes that arise from multiple minima. Experiments show that the architecture maintains the single-critical-point property and succeeds on tasks where prior methods fail.

Core claim

The paper claims that PolarNet, through its specific architectural constraints, ensures the modeled function has precisely one critical point, while also satisfying the mathematical conditions of properness and universality required for Lyapunov functions, thereby enabling reliable use in neural Lyapunov control without the training instabilities caused by multiple minima in conventional networks.

What carries the argument

PolarNet, a neural network architecture engineered to possess exactly one critical point.

If this is right

Existing neural Lyapunov control methods can use PolarNet as a direct replacement to sidestep training failures linked to multiple critical points.
The resulting Lyapunov functions maintain a single minimum, which supports consistent stability certification.
Theoretical guarantees on properness and universality hold for the functions produced by the architecture.
Numerical tests confirm avoidance of the identified training difficulties across several control problems.

Where Pith is reading between the lines

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

The single-critical-point design could be adapted to other function approximation tasks that require unique minima, such as certain optimization or energy landscapes.
Testing PolarNet on higher-dimensional or partially observed systems would reveal whether the guarantees scale beyond the reported cases.
Similar architectural constraints might be developed for other certificates like barrier functions in safety-critical control.

Load-bearing premise

That building a network to have only one critical point is sufficient to produce a valid Lyapunov function that works for proving stability across arbitrary system dynamics.

What would settle it

A numerical experiment in which a trained PolarNet produces a function with more than one critical point, or where the function fails to satisfy the Lyapunov decrease condition along system trajectories despite the single-point architecture.

Figures

Figures reproduced from arXiv: 2605.09546 by Hefu Ye, Jiaxin Cheng, Yicong Zhou, Yuan Zhong.

**Figure 2.** Figure 2: Graphical illustration of Theorem 1: The case of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: Results of fitting four different V (x) : R 2 → R. In (a), V is a single pole function. In (b), (c) and (d), V has multiple local minima and thus is not a single pole function. Note that a network that guarantees to be a single pole function should fail to faithfully represent V in cases (b), (c), (d). We visualize R 2 → R as contour lines for visual clarity, using the same scheme as in [PITH_FULL_IMAGE:f… view at source ↗

**Figure 4.** Figure 4: As shown in [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

**Figure 4.** Figure 4: Controller synthesis result with different network architectures: (a1), (b1): Lyapunov [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

**Figure 5.** Figure 5: Controller synthesis result with different network architectures: (a1), (b1): Lyapunov [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗

read the original abstract

Learning control strategies with provable stability guarantees continues to be a challenging problem. In this work, we examine a family of training-time behaviors exhibited by existing neural Lyapunov control methods under specific conditions, which can hinder the synthesis of a provably stable controller. We identify the root cause as the lack of neural network architectural guarantees on the learned Lyapunov function, and propose PolarNet, a network architecture that provably addresses these issues by structurally guarantee to have a single critical point. We provide theoretical guarantee regarding the properness and universality of PolarNet for modeling Lyapunov functions, and show that using it as a drop-in replacement in existing neural Lyapunov control methods can effectively circumvent particular difficulties in training. We conduct a set of numerical experiments to verify that PolarNet consistently maintains a single critical point and, when used as a drop-in replacement in existing neural Lyapunov control methods, successfully avoids training failures caused by the lack of architectural guarantees. The code of this paper is available at https://github.com/23-zy/PolarNet.

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.

Desk Editor's Note private letter to a colleague

PolarNet bakes a single critical point at the origin into the network architecture for Lyapunov modeling, with proofs of properness and universality that let it replace standard nets without some common training breakdowns.

read the letter

PolarNet is a network whose output is forced to have only one critical point, at the origin, plus proofs that the resulting function is proper and can approximate the right class of Lyapunov candidates. This is the core new piece: an architectural guarantee rather than another loss term or initialization heuristic to keep the learned V positive definite and radially unbounded. The paper shows it drops into existing neural Lyapunov pipelines and removes certain failure modes that come from multiple minima during training. The public code is a plus for reproducing the single-critical-point checks. The experiments confirm the architecture behaves as claimed on the tested examples and that training succeeds where prior nets sometimes do not. That is useful for anyone who has watched a learned Lyapunov function develop extra local minima and break the stability certificate. The soft spots are modest but real. The single-minimum property is necessary for a valid Lyapunov function but still leaves the negative-derivative condition to be enforced by the optimizer, so the method does not remove all training risk. The numerical cases are limited in dimension and complexity, and the universality claim likely rests on smoothness or growth assumptions that may not cover every practical system. No load-bearing circularity or hidden self-reference appears in the central argument. This paper is for researchers who build learned controllers with neural certificates and have hit the multiple-minima problem. Readers already working in that niche will get immediate practical value from the architecture and the code. It is focused enough and grounded enough to deserve a serious referee, even if the experiments stay at the proof-of-concept level. I would send it out for review.

Referee Report

2 major / 2 minor

Summary. The paper proposes PolarNet, a neural network architecture for modeling Lyapunov functions that structurally enforces a single critical point at the origin. It claims theoretical guarantees of properness and universality within the class of functions suitable for Lyapunov analysis, and demonstrates via numerical experiments that the architecture serves as a drop-in replacement in existing neural Lyapunov control methods to avoid specific training pathologies such as multiple minima.

Significance. If the architectural guarantees and proofs hold, PolarNet could meaningfully improve reliability in neural Lyapunov control by eliminating a common source of training instability without requiring changes to the loss or optimization procedure. The open-source code release supports reproducibility and allows direct testing of the single-critical-point property.

major comments (2)

[§3] §3 (Theoretical guarantees): The universality claim for PolarNet requires explicit statement of the function class and any restrictions on the system dynamics; the abstract states that single-minimum plus properness suffices for a valid Lyapunov function, but it is unclear whether this holds without additional assumptions on radial unboundedness or the form of the vector field.
[Experimental results] Experimental results (numerical verification section): The experiments confirm single-critical-point behavior for PolarNet but do not report quantitative comparisons (e.g., success rate or failure frequency) against baseline neural Lyapunov methods across the tested systems; without these metrics the claim that PolarNet 'successfully avoids training failures' remains qualitative.

minor comments (2)

[Abstract] Abstract: 'structurally guarantee' should read 'structurally guarantees'.
[§2] Notation: The definition of the PolarNet layers and the precise location of the enforced critical point should be stated with an equation number in the main text for easy reference.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback. We address each major comment below with clarifications and indicate the revisions we will incorporate to improve the manuscript.

read point-by-point responses

Referee: [§3] §3 (Theoretical guarantees): The universality claim for PolarNet requires explicit statement of the function class and any restrictions on the system dynamics; the abstract states that single-minimum plus properness suffices for a valid Lyapunov function, but it is unclear whether this holds without additional assumptions on radial unboundedness or the form of the vector field.

Authors: We agree that greater precision is needed. In the revised manuscript, we will explicitly state the function class: the set of C^1 functions f: R^n -> R that are positive definite, proper (i.e., radially unbounded, lim_{||x||->infty} f(x) = infty), and possess a unique critical point at the origin. Properness is the standard mathematical term for radial unboundedness and is already proven for PolarNet; we will add a sentence clarifying this equivalence. Our universality result shows that PolarNet can approximate any function in this class to arbitrary accuracy (in the C^1 topology on compact sets), independent of any particular vector field. When PolarNet is used as a Lyapunov function for a given system, the negative-definiteness of the Lie derivative is enforced separately via the training loss or verification step, as is standard in neural Lyapunov control. We will revise Section 3 and the abstract accordingly to remove any ambiguity. revision: yes
Referee: [Experimental results] Experimental results (numerical verification section): The experiments confirm single-critical-point behavior for PolarNet but do not report quantitative comparisons (e.g., success rate or failure frequency) against baseline neural Lyapunov methods across the tested systems; without these metrics the claim that PolarNet 'successfully avoids training failures' remains qualitative.

Authors: We acknowledge that the current experiments are primarily qualitative demonstrations of the single-critical-point property and avoidance of specific pathologies. To strengthen the empirical claims, we will add quantitative metrics in the revised numerical verification section: specifically, success rates (fraction of independent training runs that converge to a valid Lyapunov function satisfying the stability certificate) and failure frequencies, computed over 20 random seeds for both PolarNet and the baseline architectures on each of the tested systems. These results will be presented in a new table or bar chart to support the statement that PolarNet avoids training failures more reliably. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines PolarNet as an explicit architectural construction whose single-critical-point property at the origin follows directly from the polar-coordinate parameterization and radial basis structure; this is not a fitted or data-dependent claim but a structural invariant proven by direct differentiation of the network output. The subsequent proofs of properness (radial unboundedness) and universality (density in the class of positive-definite functions) are standard analytic arguments that invoke only the architectural form and classical Lyapunov conditions, without any reduction to self-citations, ansatzes imported from prior author work, or renaming of empirical patterns. No equation equates a learned quantity to a prediction of itself, and the central guarantee is independent of any training data or fitted parameters. The derivation chain is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

PolarNet is presented as a new architectural construction whose single-critical-point property is claimed to be structural. No free parameters are mentioned in the abstract. The universality and properness claims rest on standard neural-network approximation results plus the new architectural constraint.

axioms (1)

standard math Standard results on neural network approximation power and Lyapunov function existence for stable systems.
Invoked to support universality claim for modeling Lyapunov functions.

invented entities (1)

PolarNet architecture no independent evidence
purpose: Neural network with enforced single critical point for Lyapunov modeling.
New construction introduced in the paper; no independent evidence outside the architectural definition itself.

pith-pipeline@v0.9.0 · 5479 in / 1209 out tokens · 39015 ms · 2026-05-12T03:51:42.434413+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes
PolarNet models a single pole function … exactly one critical point at x=0: ∇V≠0 ∀x≠0 … V=||Ψ(x)||² where Ψ is C^∞ smooth, invertible and zero-crossing … Theorem 2: If Ψ … then ||Ψ(x)||² is a single pole function.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear
Proof of Theorem 1 … Morse lemma … gradient flow … diffeomorphism between sublevel sets and spheres … Alexander duality not used.

Reference graph

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