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arxiv: 2605.05530 · v1 · submitted 2026-05-07 · 💻 cs.LG

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Energy Generative Modeling: A Lyapunov-based Energy Matching Perspective

Yixuan Wang , Wenqian Xue , Warren E. Dixon
Authors on Pith no claims yet

Pith reviewed 2026-05-09 15:58 UTC · model grok-4.3

classification 💻 cs.LG
keywords energy-based generative modelsLyapunov functionsWasserstein spaceLangevin samplingdensity transportnonlinear controlGibbs measure
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The pith

Static scalar energies unify training and sampling in generative models as controlled density transport on Wasserstein space with KL divergence as Lyapunov function.

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

Generative models based on static scalar energy functions treat sample generation as gradient flow driven by a time-independent potential. The paper unifies training and sampling by framing both as density transport dynamics on the Wasserstein space, formulated as a nonlinear control problem. In this setup the Kullback-Leibler divergence to the target Gibbs measure acts as a Lyapunov function that decreases along the closed-loop trajectories. This perspective yields a finite-step stopping criterion for Langevin sampling and shows that deterministic gradient flow admits no such certificate. It also proves that sums of trained energies preserve the invariant measure and the Lyapunov property.

Core claim

We unify the training and sampling phases of this paradigm within a single framework: density transport on the Wasserstein space, cast as a nonlinear control problem in which the Kullback Leibler (KL) divergence serves as a Lyapunov function. Training and sampling are then two instances of this same master dynamics, differing only in initial condition. Within this autonomous framework we develop two analytic results. First, since the Lyapunov certificate is asymptotic, we derive a finite step stopping criterion for Langevin sampling and prove that no Lyapunov certificate exists for the deterministic gradient flow on the same energy landscape. Second, the reformulation brings the toolkit of 1

What carries the argument

Density transport on Wasserstein space formulated as a nonlinear control problem in which the KL divergence to the target Gibbs measure serves as the Lyapunov function for dynamics driven by the gradient of a static scalar energy.

Load-bearing premise

The KL divergence between the evolving density and the target Gibbs measure serves as a valid, strictly decreasing Lyapunov function for the controlled density transport dynamics on Wasserstein space when the control is given by the gradient of a static scalar energy.

What would settle it

A numerical check on a multimodal target showing that KL divergence fails to decrease monotonically under the proposed energy-gradient control, or that a Lyapunov function can be found for deterministic gradient flow.

Figures

Figures reproduced from arXiv: 2605.05530 by Warren E. Dixon, Wenqian Xue, Yixuan Wang.

Figure 1
Figure 1. Figure 1: Trajectory snapshots on the eight Gaussian targets with sampling step view at source ↗
Figure 2
Figure 2. Figure 2: Compositional generation by additive energy operations. From left to right: expert view at source ↗
read the original abstract

Generative models based on static scalar energy functions represent an emerging paradigm in which a single time independent potential drives sample generation through its gradient field, eliminating the need for time conditioning entirely. We unify the training and sampling phases of this paradigm, conventionally treated as separate procedures, within a single framework: density transport on the Wasserstein space, cast as a nonlinear control problem in which the Kullback Leibler (KL) divergence serves as a Lyapunov function. Training and sampling are then two instances of this same master dynamics, differing only in initial condition. Within this autonomous framework we develop two analytic results. First, since the Lyapunov certificate is asymptotic, we derive a finite step stopping criterion for Langevin sampling and prove that no Lyapunov certificate exists for the deterministic gradient flow on the same energy landscape. Second, the reformulation brings the toolkit of nonlinear control theory to bear on static scalar energy generative modeling, that is, we show that additive composition of trained scalar energies retains an explicit Gibbs invariant measure and inherits the closed-loop Lyapunov certificate. Beyond these immediate results, this reformulation bridges static scalar energy generative models with the full toolkit of nonlinear control theory, opening the door to barrier functions for constrained generation and contraction metrics for accelerated sampling. Experiments on synthetic distributions validate the theoretical predictions.

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 / 2 minor

Summary. The manuscript unifies training and sampling in static scalar energy generative models by modeling density transport on Wasserstein space as a nonlinear control problem, using KL divergence as a Lyapunov function. It presents two main analytic results: a finite-step stopping criterion for Langevin sampling and a proof that no Lyapunov certificate exists for the deterministic gradient flow on the same landscape. Additionally, it shows that additive composition of trained energies preserves the Gibbs invariant measure and the Lyapunov certificate. Synthetic experiments validate the theoretical findings.

Significance. This reformulation offers a promising connection to nonlinear control theory, which could enable new methods for constrained generation and accelerated sampling. The analytic results on stopping criteria and energy composition are potentially significant for practical implementation of energy-based models. The standard calculations for the stochastic case align with known Fokker-Planck dynamics, and the distinction for the deterministic case is well-motivated. If the derivations are complete, this work strengthens the theoretical foundation of the paradigm.

major comments (2)
  1. [First analytic result on finite stopping criterion] The finite step stopping criterion is derived from the asymptotic nature of the Lyapunov function. To ensure it is not post-hoc, the manuscript should provide the explicit mathematical form of the criterion, including any dependence on step size or energy bounds (e.g., near the relevant equation in the sampling section). This is important for the claim's practicality.
  2. [Proof of no Lyapunov certificate for deterministic gradient flow] While the sign-indefinite nature of d/dt KL under the continuity equation is shown, the stronger claim that 'no Lyapunov certificate exists' requires demonstrating that the deterministic flow fails to converge to the target measure in general. A specific counterexample or reference to stability theory would strengthen this.
minor comments (2)
  1. [Experiments section] The synthetic experiments are mentioned; adding quantitative metrics or comparisons to standard methods would enhance the validation of the theoretical predictions.
  2. [Introduction] A brief review of related work on Lyapunov functions in sampling or control in generative models would help contextualize the contribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and insightful comments on our manuscript. These suggestions will help clarify the presentation of our analytic results on the finite stopping criterion and the non-existence of a Lyapunov certificate for the deterministic flow. We address each major comment point by point below and will incorporate revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: [First analytic result on finite stopping criterion] The finite step stopping criterion is derived from the asymptotic nature of the Lyapunov function. To ensure it is not post-hoc, the manuscript should provide the explicit mathematical form of the criterion, including any dependence on step size or energy bounds (e.g., near the relevant equation in the sampling section). This is important for the claim's practicality.

    Authors: We agree that an explicit mathematical form will improve practicality and rigor. In the revised manuscript, we will add the explicit stopping criterion near the relevant equation in the sampling section. The criterion will be stated as a finite time T such that the KL divergence falls below a threshold derived from the Lyapunov decrease rate, explicitly depending on the discretization step size h and an upper bound on the energy function (ensuring the sampled measure is within epsilon of the target Gibbs measure). This makes the result directly usable without appearing post-hoc. revision: yes

  2. Referee: [Proof of no Lyapunov certificate for deterministic gradient flow] While the sign-indefinite nature of d/dt KL under the continuity equation is shown, the stronger claim that 'no Lyapunov certificate exists' requires demonstrating that the deterministic flow fails to converge to the target measure in general. A specific counterexample or reference to stability theory would strengthen this.

    Authors: We thank the referee for highlighting this distinction. Our proof already shows that d/dt KL is sign-indefinite along the deterministic continuity equation, implying KL itself cannot serve as a Lyapunov function. To address the stronger claim of no Lyapunov certificate existing in general, we will revise the manuscript to include a reference to stability theory for Wasserstein gradient flows and add a simple counterexample: an energy landscape (e.g., a non-convex potential) where the deterministic flow from a specific initial measure does not converge to the target Gibbs measure. This will be placed in the deterministic flow section. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation applies standard nonlinear control theory and Fokker-Planck analysis to Wasserstein-space density transport under static energy gradient control. The KL divergence is shown to be a Lyapunov function via the explicit computation d/dt KL(p||p*) = -∫ p |∇log(p/p*)|^2 ≤ 0 for the stochastic case and the corresponding sign-indefinite expression for the deterministic continuity equation; both follow directly from the underlying PDEs without reference to fitted parameters or self-referential definitions inside the paper. The finite stopping criterion is a standard consequence of asymptotic stability, the non-existence result for deterministic flow is obtained by direct comparison of the two dynamics, and the additive composition result is an immediate algebraic property of the Gibbs measure exp(-(E1+E2)). No load-bearing step reduces by construction to an input, a self-citation chain, or a renamed empirical pattern; the framework remains self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on the standard assumption that the Wasserstein gradient flow of the KL divergence is well-defined and that the energy function induces a valid control input; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption KL divergence serves as a Lyapunov function for the controlled density transport on Wasserstein space
    Invoked to unify training and sampling and to derive the stopping criterion and composition result.

pith-pipeline@v0.9.0 · 5524 in / 1447 out tokens · 30099 ms · 2026-05-09T15:58:43.216760+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

52 extracted references · 12 canonical work pages · 2 internal anchors

  1. [1]

    Advances in Neural Information Processing Systems , volume=

    Denoising diffusion probabilistic models , author=. Advances in Neural Information Processing Systems , volume=

  2. [2]

    International Conference on Learning Representations , year=

    Score-based generative modeling through stochastic differential equations , author=. International Conference on Learning Representations , year=

  3. [3]

    International Conference on Learning Representations , year=

    Flow matching for generative modeling , author=. International Conference on Learning Representations , year=

  4. [4]

    International Conference on Learning Representations , year=

    Flow straight and fast: Learning to generate and transfer data with rectified flow , author=. International Conference on Learning Representations , year=

  5. [5]

    Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions.arXiv preprint arXiv:2209.11215, 2022c

    Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions , author=. arXiv preprint arXiv:2209.11215 , year=

    work page arXiv

  6. [6]

    Nearlyd-linear conver- gence bounds for diffusion models via stochastic localization.arXiv preprint arXiv:2308.03686,

    Nearly d -linear convergence bounds for diffusion models via stochastic localization , author=. arXiv preprint arXiv:2308.03686 , year=

    work page arXiv

  7. [7]

    arXiv preprint arXiv:2306.09251 , year=

    Towards faster non-asymptotic convergence for diffusion-based generative models , author=. arXiv preprint arXiv:2306.09251 , year=

    work page arXiv

  8. [8]

    arXiv preprint arXiv:2504.10612 , year=

    Energy Matching: Unifying flow matching and energy-based models for generative modeling , author=. arXiv preprint arXiv:2504.10612 , year=

    work page arXiv

  9. [9]

    and Du, Y

    Equilibrium Matching , author=. arXiv preprint arXiv:2510.02300 , year=

    work page arXiv

  10. [10]

    The variational formulation of the

    Jordan, Richard and Kinderlehrer, David and Otto, Felix , journal=. The variational formulation of the

  11. [11]

    Optimal Transport: Old and New , author=

  12. [12]

    Diffusions hypercontractives , author=. S. 1985 , publisher=

    1985

  13. [13]

    Analysis and Geometry of

    Bakry, Dominique and Gentil, Ivan and Ledoux, Michel , publisher=. Analysis and Geometry of

  14. [14]

    Risken, Hannes , publisher=. The

  15. [15]

    Introduction to Nonparametric Estimation , author=

  16. [16]

    Logarithmic

    Holley, Richard and Stroock, Daniel , journal=. Logarithmic

  17. [17]

    Advances in Neural Information Processing Systems , volume=

    Compositional visual generation with energy based models , author=. Advances in Neural Information Processing Systems , volume=

  18. [18]

    Reduce, Reuse, Recycle: Compositional generation with energy-based diffusion models and

    Du, Yilun and Durkan, Conor and Strudel, Robin and Tenenbaum, Joshua B and Dieleman, Sander and Fergus, Rob and Sohl-Dickstein, Jascha and Doucet, Arnaud and Grathwohl, Will Sussman , booktitle=. Reduce, Reuse, Recycle: Compositional generation with energy-based diffusion models and

  19. [19]

    Underdamped

    Cheng, Xiang and Bartlett, Naomi S and Jordan, Michael I and Bartlett, Peter L , journal=. Underdamped

  20. [20]

    arXiv preprint arXiv:2402.05774 , year=

    Stable Autonomous Flow Matching , author=. arXiv preprint arXiv:2402.05774 , year=

    work page arXiv

  21. [21]

    Linear Algebra and its Applications , volume=

    Dynamical systems that sort lists, diagonalize matrices, and solve linear programming problems , author=. Linear Algebra and its Applications , volume=

  22. [22]

    Advances in Neural Information Processing Systems , volume=

    Generative modeling by estimating gradients of the data distribution , author=. Advances in Neural Information Processing Systems , volume=

  23. [23]

    Predicting Structured Data , volume=

    A tutorial on energy-based learning , author=. Predicting Structured Data , volume=

  24. [24]

    Advances in Neural Information Processing Systems , volume=

    Energy-based out-of-distribution detection , author=. Advances in Neural Information Processing Systems , volume=

  25. [25]

    International Conference on Machine Learning , pages=

    Deep unsupervised learning using nonequilibrium thermodynamics , author=. International Conference on Machine Learning , pages=

  26. [26]

    Stochastic Interpolants: A Unifying Framework for Flows and Diffusions

    Stochastic interpolants: A unifying framework for flows and diffusions , author=. arXiv preprint arXiv:2303.08797 , year=

    work page internal anchor Pith review arXiv

  27. [27]

    Transactions on Machine Learning Research , year=

    Improving and generalizing flow-based generative models with minibatch optimal transport , author=. Transactions on Machine Learning Research , year=

  28. [28]

    European Conference on Computer Vision , pages=

    Sit: Exploring flow and diffusion-based generative models with scalable interpolant transformers , author=. European Conference on Computer Vision , pages=

  29. [29]

    Advances in Neural Information Processing Systems , volume=

    Convergence for score-based generative modeling with polynomial complexity , author=. Advances in Neural Information Processing Systems , volume=

  30. [30]

    arXiv preprint arXiv:2211.01916 , year=

    Improved analysis of score-based generative modeling: User-friendly bounds under minimal smoothness assumptions , author=. arXiv preprint arXiv:2211.01916 , year=

    work page arXiv

  31. [31]

    How to train your energy-based models.arXiv preprint arXiv:2101.03288,

    How to train your energy-based models , author=. arXiv preprint arXiv:2101.03288 , year=

    work page arXiv

  32. [32]

    Neural Computation , volume=

    Training products of experts by minimizing contrastive divergence , author=. Neural Computation , volume=

  33. [33]

    Nonlinear Systems , author=

  34. [34]

    Mathematical Control Theory: Deterministic Finite Dimensional Systems , author=

  35. [35]

    Advances in Neural Information Processing Systems , volume=

    Implicit generation and modeling with energy based models , author=. Advances in Neural Information Processing Systems , volume=

  36. [36]

    International Conference on Learning Representations , year=

    Your classifier is secretly an energy based model and you should treat it like one , author=. International Conference on Learning Representations , year=

  37. [37]

    Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics , pages=

    Noise-contrastive estimation: A new estimation principle for unnormalized statistical models , author=. Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics , pages=

  38. [38]

    International Conference on Machine Learning , pages=

    Improved denoising diffusion probabilistic models , author=. International Conference on Machine Learning , pages=

  39. [39]

    Diffusion models beat

    Dhariwal, Prafulla and Nichol, Alexander , journal=. Diffusion models beat

  40. [40]

    European Conference on Computer Vision , pages=

    Compositional visual generation with composable diffusion models , author=. European Conference on Computer Vision , pages=

  41. [41]

    Score as action: Fine-tuning diffusion generative models by continuous-time reinforcement learning.arXiv preprint arXiv:2502.01819, 2025

    Score as Action: Fine-Tuning Diffusion Generative Models by Continuous-Time Reinforcement Learning , author=. arXiv preprint arXiv:2502.01819 , year=

    work page arXiv

  42. [42]

    Neural Computation , volume=

    Natural gradient works efficiently in learning , author=. Neural Computation , volume=

  43. [43]

    Riemann manifold

    Girolami, Mark and Calderhead, Ben , journal=. Riemann manifold

  44. [44]

    IEEE Transactions on Automatic Control , volume=

    Control barrier function based quadratic programs for safety critical systems , author=. IEEE Transactions on Automatic Control , volume=

  45. [45]

    Sampling from a log-concave distribution with compact support with proximal

    Brosse, Nicolas and Durmus, Alain and Moulines,. Sampling from a log-concave distribution with compact support with proximal. Statistics and Computing , volume=

  46. [46]

    Is there an analog of

    Ma, Yi-An and Chatterji, Niladri S and Cheng, Xiang and Flammarion, Nicolas and Bartlett, Peter L and Jordan, Michael I , journal=. Is there an analog of

  47. [47]

    Generative Modeling via Drifting

    Generative Modeling via Drifting , author=. arXiv preprint arXiv:2602.04770 , year=

    work page internal anchor Pith review arXiv

  48. [48]

    Gradient flow drifting: Generative modeling via wasserstein gradient flows of kde-approximated divergences.arXiv preprint arXiv:2603.10592,

    Gradient Flow Drifting: Generative Modeling via Wasserstein Gradient Flows of KDE-Approximated Divergences , author=. arXiv preprint arXiv:2603.10592 , year=

    work page arXiv

  49. [49]

    Estimation of Non-Normalized Statistical Models by Score Matching , journal =

    Aapo Hyv. Estimation of Non-Normalized Statistical Models by Score Matching , journal =. 2005 , volume =

    2005

  50. [50]

    A Connection Between Score Matching and Denoising Autoencoders , volume =

    Vincent, Pascal , year =. A Connection Between Score Matching and Denoising Autoencoders , volume =. Neural Computation , doi =

  51. [51]

    Hoti, Fabian , year =

  52. [52]

    2020 , publisher=

    The theory of Lie derivatives and its applications , author=. 2020 , publisher=

    2020