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arxiv: 2606.23550 · v2 · pith:NLEP3UHBnew · submitted 2026-06-22 · ❄️ cond-mat.dis-nn · cs.LG

Approximating velocity fields with planted attractors via Neural-ODEs for classification purposes

Feliciano Giuseppe Pacifico , Duccio Fanelli , Lorenzo Buffoni , Lorenzo Chicchi , Diego Febbe , Raffaele Marino This is my paper

Pith reviewed 2026-06-26 05:46 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn cs.LG
keywords Neural ODEsclassificationattractorsvelocity fieldsbasins of attractiondynamical systemsmachine learning
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The pith

Neural ODEs classify data by planting equilibrium points that define class-specific basins of attraction.

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

The paper demonstrates that Neural ODEs can perform classification by pre-selecting equilibrium points as class labels and training the velocity field to route inputs toward those points. Each data sample enters the system as an initial condition and follows the learned flow until it reaches its assigned attractor. The approach relies on the network's ability to approximate velocity fields that carve out separate basins without significant overlap. This turns the classification problem into one of shaping a dynamical landscape around fixed targets.

Core claim

Neural ODEs equipped with a curated collection of equilibrium points have been successfully employed for classification tasks. The planted attractors serve as indicators for the target classes, while the velocity field leveraging the universal approximation capabilities of the architecture shapes the dynamical landscape. This process defines the basins of attraction of the trained model, effectively directing each input (provided as an initial condition) toward its corresponding destination target.

What carries the argument

Planted equilibrium points that act as class indicators while the Neural ODE velocity field is trained to create matching basins of attraction.

If this is right

  • Classification reduces to determining which planted attractor an input trajectory reaches.
  • The universal approximation property of the Neural ODE allows the velocity field to be shaped arbitrarily around the fixed points.
  • The resulting dynamical system assigns every initial condition to exactly one class via its basin.
  • No explicit decision boundary is needed; the flow itself performs the separation.

Where Pith is reading between the lines

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

  • The method could be tested on datasets where the natural geometry already contains multiple stable fixed points.
  • Performance may degrade if the planted points are placed too close together in high-dimensional input space.
  • The same construction might apply to regression tasks by planting a continuum of attractors instead of discrete points.

Load-bearing premise

A suitable collection of equilibrium points can be chosen in advance and training will produce a velocity field whose basins of attraction align with the desired class labels without significant overlap or misclassification.

What would settle it

A test set in which a substantial fraction of inputs from one class converge to the attractor of a different class after training.

Figures

Figures reproduced from arXiv: 2606.23550 by Diego Febbe, Duccio Fanelli, Feliciano Giuseppe Pacifico, Lorenzo Buffoni, Lorenzo Chicchi, Raffaele Marino.

Figure 1
Figure 1. Figure 1: FIG. 1: Two-spirals dynamical classification with planted equilibria (ConsNODEs). ( [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Accuracy of ConsNODEs on Fashion-MNIST as a function of the integration time. The validation [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: CIFAR-10 accuracy learning curves (mean [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Projected trajectories of the coupled model (22) in the [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Diagnostics for the coupled system (22) at small [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
read the original abstract

In this work, Neural ODEs equipped with a curated collection of equilibrium points have been successfully employed for classification tasks. The planted attractors serve as indicators for the target classes, while the velocity field leveraging the universal approximation capabilities of the architecture shapes the dynamical landscape. This process defines the basins of attraction of the trained model, effectively directing each input (provided as an initial condition) toward its corresponding destination target.

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

2 major / 0 minor

Summary. The manuscript claims that Neural ODEs equipped with a curated collection of pre-planted equilibrium points (attractors) can be successfully used for classification tasks. The planted attractors indicate target classes, and the trained velocity field is asserted to shape the dynamical landscape so that basins of attraction direct each input (as an initial condition) to its correct class label, leveraging the universal approximation property of the architecture.

Significance. If the central claim were substantiated, the approach would offer an interpretable dynamical-systems framing for classification in which class labels correspond to basins of attraction around planted equilibria. However, the manuscript supplies no equations, training procedure, loss function, dataset details, performance metrics, baselines, or error analysis, so no assessment of significance is possible.

major comments (2)
  1. [Abstract] Abstract: the claim that the method has been 'successfully employed' for classification is made without any derivation, model equations, training algorithm, convergence argument, or experimental results, so the central claim that training produces non-overlapping basins aligned with class labels cannot be evaluated.
  2. [Abstract] Abstract: the appeal to 'universal approximation capabilities' guarantees existence of some velocity field but does not address whether gradient-based training on finite data will recover a field whose flow maps every point to the correct planted attractor without basin overlap or misrouting, which is the load-bearing requirement for the classification claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their comments. The current manuscript is a brief conceptual outline relying on the universal approximation property of Neural ODEs, but we agree that it lacks the concrete equations, training details, and empirical validation needed to substantiate the classification claim. We will revise the manuscript to supply these elements.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that the method has been 'successfully employed' for classification is made without any derivation, model equations, training algorithm, convergence argument, or experimental results, so the central claim that training produces non-overlapping basins aligned with class labels cannot be evaluated.

    Authors: We accept that the abstract's phrasing requires supporting material. The revised manuscript will include the explicit Neural-ODE vector field with planted equilibria, the loss function that penalizes trajectories ending in the wrong basin, the gradient-based training procedure, and experimental results on standard datasets demonstrating the resulting basins. revision: yes

  2. Referee: [Abstract] Abstract: the appeal to 'universal approximation capabilities' guarantees existence of some velocity field but does not address whether gradient-based training on finite data will recover a field whose flow maps every point to the correct planted attractor without basin overlap or misrouting, which is the load-bearing requirement for the classification claim.

    Authors: The universal approximation result only guarantees existence; we do not claim a general convergence proof for gradient descent on finite samples. The revised version will add a discussion of the concrete loss and optimization choices used in practice, together with empirical checks for basin overlap on the datasets considered. A rigorous guarantee that training always succeeds without misrouting remains an open theoretical question and is not asserted in the manuscript. revision: partial

Circularity Check

0 steps flagged

No derivation chain or equations presented; circularity cannot be assessed

full rationale

The provided abstract and description contain no equations, derivations, self-citations, or load-bearing steps that could reduce to inputs by construction. The claim relies on universal approximation of Neural ODEs and training to shape basins, which are standard external capabilities not shown to be internally circular. No specific reduction (e.g., fitted parameter renamed as prediction) is visible, so the finding is no significant circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The abstract invokes the universal approximation property of neural networks to justify shaping arbitrary velocity fields and assumes that equilibrium points can be curated to serve as class indicators. No explicit free parameters or invented entities beyond the planted attractors are named.

axioms (1)
  • domain assumption Neural networks possess universal approximation capabilities
    Invoked in the abstract to support that the velocity field can shape the dynamical landscape as needed.
invented entities (1)
  • planted attractors no independent evidence
    purpose: Serve as indicators for the target classes in the classification task
    Introduced as curated equilibrium points that define the basins of attraction

pith-pipeline@v0.9.1-grok · 5606 in / 1108 out tokens · 24575 ms · 2026-06-26T05:46:36.968337+00:00 · methodology

discussion (0)

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Reference graph

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