pith. sign in

arxiv: 2607.02349 · v1 · pith:R2Z3RRUJnew · submitted 2026-07-02 · 🧮 math.NA · cs.NA· math.PR

A PDE-Based Framework for Generative Modeling Beyond Classical Score-Based Diffusion

Horacio Tettamanti , Michael Herty This is my paper

Pith reviewed 2026-07-03 07:29 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath.PR
keywords generative modelingnonlinear Fokker-Planck equationreverse-time PDEOrnstein-Uhlenbeck dynamicscondensation phenomenascore-based diffusioninteracting particle systemsdensity filtering
0
0 comments X

The pith

A nonlinear modification of Ornstein-Uhlenbeck dynamics yields a stabilized reverse-time PDE that reconstructs initial distributions from asymptotic condensed states.

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

The paper develops an alternative to score-based diffusion by replacing the standard linear drift with a nonlinear term in the Ornstein-Uhlenbeck process. This produces a forward evolution whose mean-field description is a nonlinear Fokker-Planck equation that loses L2 regularity in finite time under suitable parameters and large mass. From the resulting asymptotic state, the authors derive a stabilized reverse-time PDE whose solution recovers the original distribution. The construction is accompanied by consistent numerical schemes that are tested on density-filtering tasks in one and two dimensions. The approach matters because it removes the need to estimate a score function and instead relies on the condensation property of the nonlinear dynamics.

Core claim

By introducing a nonlinear modification of the classical Ornstein-Uhlenbeck dynamics that admits both a particle-system and a mean-field nonlinear Fokker-Planck description with superlinear drift, the forward process reaches an asymptotic state from which a stabilized reverse-time partial differential equation reconstructs the initial distribution, thereby extending the generative paradigm beyond the classical score-based framework.

What carries the argument

The stabilized reverse-time partial differential equation obtained by reversing the nonlinear Fokker-Planck equation with superlinear drift term.

If this is right

  • For suitable parameters and sufficiently large initial mass the forward dynamics loses L2 regularity in finite time through condensation.
  • Numerical discretizations of the forward and reverse processes accurately reproduce the continuous model's asymptotic behavior and recover the initial distribution.
  • Iterated application of the generative process performs density filtering.
  • The method is demonstrated by experiments in one and two spatial dimensions.

Where Pith is reading between the lines

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

  • The condensation mechanism may offer an alternative route to stable high-dimensional sampling that avoids explicit score estimation.
  • Similar nonlinear drifts could be substituted to produce different condensation rates or target measures while preserving the reverse reconstruction property.
  • The framework invites comparison with other mean-field limits of interacting particles to identify which nonlinearities permit reliable reverse-time recovery.

Load-bearing premise

The interacting particle system admits a mean-field limit given exactly by the nonlinear Fokker-Planck equation with superlinear drift term.

What would settle it

A concrete counterexample in which the proposed reverse-time PDE produces a density that differs substantially from the known initial distribution when the forward process is run to its asymptotic state.

Figures

Figures reproduced from arXiv: 2607.02349 by Horacio Tettamanti, Michael Herty.

Figure 1
Figure 1. Figure 1: Generative process of the numerical scheme with [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Generative process of the numerical scheme with [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Reconstruction of a two-dimensional distribution via forward and backward diffusion dynamics. [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Smoothing effect of the generative process on a noisy distribution. In (a) we show the initial distribution [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Generative process acting as a filtering mechanism on a noisy two-dimensional distribution. (a) Initial [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Generative application for Image smoothing. In (a) we show the original image, a noisy image after [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Particles initially sampled from a ring distribution (top row) spread under the forward dynamics. The [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Marginal distributions obtained at the end of the forward and reverse processes together with the [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: By modifying the prior distribution, it is possible to reconstruct regions of a density where data is [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: For different values of A, we evaluate the relative L 1 error between the reconstructed and analytical distributions. We observe that there exists an optimal choice of A for which the reconstruction error is minimized and the original distribution is best recovered. and define the missing region as Igap = n (x, y) ∈ Ω : p x 2 + y 2 < 0.85o . We sample Np = 105 particles and evolve the system up to final t… view at source ↗
read the original abstract

We introduce an alternative generative framework based on a nonlinear modification of the classical Ornstein--Uhlenbeck dynamics. The proposed dynamics admits both a microscopic description through an interacting particle system and, in the mean-field limit, a macroscopic formulation given by a nonlinear Fokker--Planck equation with a superlinear drift term. We show that, for suitable choices of the model parameters and sufficiently large initial mass, the forward dynamics exhibits condensation phenomena by proving the loss of $L^2$ regularity of the solution in finite time. Building upon this formulation, we derive a stabilized reverse-time partial differential equation that reconstructs the initial distribution from the asymptotic state of the forward dynamics, thereby extending the generative paradigm beyond the classical score-based framework. Furthermore, we introduce numerical discretizations of both the forward and reverse processes that accurately capture the asymptotic behavior of the continuous model while successfully reconstructing the initial distribution. Numerical experiments in one and two spatial dimensions validate the proposed methodology and illustrate its application to density filtering through successive iterations of the generative process.

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 / 2 minor

Summary. The paper introduces a nonlinear modification of the Ornstein-Uhlenbeck process for generative modeling. It connects an interacting particle system to a macroscopic nonlinear Fokker-Planck equation with superlinear drift in the mean-field limit, proves finite-time loss of L² regularity (condensation) for large initial mass, derives a stabilized reverse-time PDE that reconstructs the initial distribution from the asymptotic condensed state, and presents numerical discretizations validated in 1D and 2D experiments for density reconstruction and filtering.

Significance. If the mean-field limit and reverse PDE derivation hold rigorously, the framework provides a non-score-based generative paradigm grounded in condensation phenomena, with potential for new reconstruction methods; the numerical validation and explicit connection between microscopic and macroscopic descriptions are strengths.

major comments (1)
  1. [Abstract and mean-field limit derivation] The central reconstruction claim relies on using the nonlinear Fokker-Planck equation (with superlinear drift) as the macroscopic description for deriving the stabilized reverse-time PDE. However, the mean-field limit from the interacting particle system is invoked without justification under the condensation regime: the paper proves loss of L² regularity in finite time for large initial mass, but standard mean-field arguments (propagation of chaos, tightness) require uniform integrability or moment bounds that fail precisely when L² regularity collapses. No alternative verification (e.g., relative entropy or coupling) is indicated.
minor comments (2)
  1. [Model parameters section] Clarify the precise parameter regime (nonlinearity strength, drift coefficient, initial mass threshold) under which both condensation and the mean-field limit are claimed to hold simultaneously.
  2. [Numerical experiments] The numerical experiments section should include quantitative error metrics (e.g., Wasserstein distance or L² reconstruction error) comparing the reverse process output to the true initial distribution across multiple runs.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thorough review and constructive criticism. The main concern is the justification of the mean-field limit under the condensation regime where L² regularity is lost. We address this point below and indicate the revisions we will make to clarify the scope of the mean-field derivation while preserving the core contribution of the stabilized reverse PDE at the macroscopic level.

read point-by-point responses

  1. Referee: [Abstract and mean-field limit derivation] The central reconstruction claim relies on using the nonlinear Fokker-Planck equation (with superlinear drift) as the macroscopic description for deriving the stabilized reverse-time PDE. However, the mean-field limit from the interacting particle system is invoked without justification under the condensation regime: the paper proves loss of L² regularity in finite time for large initial mass, but standard mean-field arguments (propagation of chaos, tightness) require uniform integrability or moment bounds that fail precisely when L² regularity collapses. No alternative verification (e.g., relative entropy or coupling) is indicated.

    Authors: We agree that the mean-field limit requires careful justification precisely when condensation occurs. In the manuscript the interacting-particle to nonlinear Fokker-Planck convergence is established in the pre-condensation regime where the requisite moment and integrability bounds hold; the subsequent loss of L² regularity is proved directly for the macroscopic PDE. The stabilized reverse-time PDE is then derived entirely at the level of this nonlinear Fokker-Planck equation, independent of the particle system. The particle description is used to motivate the model and to furnish the numerical scheme. We will revise the text to (i) explicitly separate the pre-condensation mean-field result from the post-condensation PDE analysis, (ii) state that the mean-field limit in the condensed regime is formal, and (iii) note that alternative justifications (relative-entropy or coupling methods) remain open questions for future work. These clarifications will be added to the introduction, the mean-field section, and the derivation of the reverse PDE. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation proceeds from forward nonlinear FP to reverse PDE without definitional reduction

full rationale

The paper introduces a nonlinear OU modification, invokes the mean-field limit to obtain the nonlinear Fokker-Planck equation, proves finite-time L2 loss (condensation), and derives a stabilized reverse-time PDE from that forward equation to reconstruct the initial distribution. No quoted step shows a fitted parameter renamed as prediction, a self-definitional loop, or a load-bearing self-citation chain that reduces the central claim to its own inputs by construction. The mean-field limit is stated as an assumption connecting microscopic to macroscopic descriptions, but the reverse-PDE derivation itself is presented as proceeding from the macroscopic equation without circular equivalence. This is the normal non-circular outcome for a derivation that remains independent of its fitted or assumed inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of the mean-field limit for the particle system and on the ability to choose model parameters that trigger finite-time condensation; these are domain-standard assumptions rather than new postulates.

free parameters (1)
  • model parameters controlling nonlinearity and drift strength
    Suitable choices required to produce condensation and loss of L2 regularity in finite time.
axioms (1)
  • domain assumption The interacting particle system converges to the nonlinear Fokker-Planck equation in the mean-field limit
    Invoked to justify the macroscopic PDE formulation used for both condensation analysis and reverse-process derivation.

pith-pipeline@v0.9.1-grok · 5711 in / 1441 out tokens · 38730 ms · 2026-07-03T07:29:40.942103+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

41 extracted references · 41 canonical work pages · 3 internal anchors

  1. [1]

    Anderson, Reverse-time diffusion equation models, Stochastic Processes and their Applications, Volume 12, Issue 3, 1982, Pages 313-326, ISSN 0304-4149

    B.D.O. Anderson, Reverse-time diffusion equation models, Stochastic Processes and their Applications, Volume 12, Issue 3, 1982, Pages 313-326, ISSN 0304-4149

    work page 1982

  2. [2]

    B. W. Silverman. Density estimation for statistics and data analysis. Routledge, 2018

    work page 2018

  3. [3]

    Bakry, I

    D. Bakry, I. Gentil, M. Ledoux, et al. Analysis and geometry of Markov diffusion operators, volume 103. Springer, 2014

    work page 2014

  4. [4]

    Baranchuk, A

    D. Baranchuk, A. Voynov, I. Rubachev, V. Khrulkov, and A. Babenko. Label-efficient semantic segmentation with diffusion models. In International Conference on Learning Representations, 2022

    work page 2022

  5. [5]

    J. A. Carrillo, K. Hopf, and J. L. Rodrigo. On the singularity formation and relaxation to equilibrium in 1D Fokker-Planck model with superlinear drift. Advances in Mathematics , 360:106883, 66, 2020

    work page 2020

  6. [6]

    H. Chen, H. Lee, J. Lu. Improved analysis of scorebased generative modeling: User-friendly bounds under minimal smoothness assumptions. In International Conference on Machine Learning, pages, 4735-4763. PMLR, 2023

    work page 2023

  7. [7]

    Calzola, G

    E. Calzola, G. Dimarco, G. Toscani, and M. Zanella. Emergence of condensation patterns in kinetic equations for opinion dynamics. Physica D: Nonlinear Phenomena , 470:134356, 2024

    work page 2024

  8. [8]

    J. A. Carrillo, M. Fornasier, J. Rosado, and G. Toscani. Asymptotic flocking dynamics for the kinetic Cucker- Smale model. SIAM J. Math. Anal. , 42(1):218–236, 2010

    work page 2010

  9. [9]

    J. A. Carrillo, M. Fornasier, G. Toscani, and F. Vecil. Particle, kinetic, and hydrodynamic models of swarming. mathematical modeling of collective behavior in socio-economic and life sciences 297336. Model. Simul. Sci. Eng. Technol., Birkhuser Boston, Inc., Boston, MA , 2010

    work page 2010

  10. [10]

    F. Cao, K. Johnston, T. Laurent, J. Le, and S. Motsch (2025). Generative diffusion models from a PDE perspective. arXiv preprint arXiv:2501.17054

    work page arXiv 2025

  11. [11]

    S. Chen, S. Chewi, J. Li, Y. Li, A. Salim, A. Zhang. Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions. In The Eleventh International Conference on Learning Representations, 2023

    work page 2023

  12. [12]

    Dieleman

    S. Dieleman. Perspectives on diffusion. https://sander.ai/2023/07/20/perspectives.html, 2023

    work page 2023

  13. [13]

    Degond and S

    P. Degond and S. Mas-Gallic. The weighted particle method for convection-diffusion equations. i. the case of an isotropic viscosity. Mathematics of computation , 53(188):485-507, 1989. 20

    work page 1989

  14. [14]

    Dhariwal, A

    P. Dhariwal, A. Nichol. Diffusion models beat gans on image synthesis. Advances in neural information processing systems, 34:878-8794, 2021. 8794, 2021

    work page 2021

  15. [15]

    Durkan, S

    C. Durkan, S. Ermon, I. Murray, and Y. Song. Maximum likelihood training of score-based diffusion models. Advances in neural information processing systems , 34:1415-1428, 2021

    work page 2021

  16. [16]

    Doucet, V

    A. Doucet, V. De Bortoli, J. Heng, and J. Thornton. Diffusion schrödinger bridge with applications to score- based generative modeling. Advances in Neural Information Processing Systems , 34:17695-17709, 2021

    work page 2021

  17. [17]

    Franceschi, L

    J. Franceschi, L. Pareschi, and M. Zanella. Superlinear drift in consensus-based optimization with condensa- tion phenomena. ESAIM: Control, Optimisation and Calculus of Variations , forthcoming, 2026

    work page 2026

  18. [18]

    Fiedler, M

    C. Fiedler, M. Herty, M. Rom, C. Segala, S. Trimpe. Reproducing kernel Hilbert spaces in the mean field limit. Kinetic and Related Models , 2023, 16(6): 850-870. doi: 10.3934/krm.2023010

    work page doi:10.3934/krm.2023010 2023

  19. [19]

    F. Golse. The mean-field limit for the dynamics of large particle systems. In Journées Équations aux Dérivées Partielles, pages Exp. No. IX, 47. Univ. Nantes, Nantes, 2003

    work page 2003

  20. [20]

    S. Gong, M. Li, J. Feng, Z. Wu, L.P. Kong. Diffuseq: Sequence to sequence text generation with diffusion models. arXiv preprint arXiv:2210.08933 , 2022

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  21. [21]

    J. Ho, A. N. Jain, and P. Abbeel. Denoising Diffusion Probabilistic Models. Advances in Neural Information Processing Systems, 33 (NeurIPS 2020), 2020

    work page 2020

  22. [22]

    J. Ho, T. Salimans, A. Gritsenko, W. Chan, M. Norouzi, and D. J. Fleet. Video diffusion models. Advances in Neural Information Processing Systems , 35:8633-8646, 2022

    work page 2022

  23. [23]

    Kaniadakis, P

    G. Kaniadakis, P. Quarati. Classical model of bosons and fermions. Phys. Rev. E , 49, 5103–5110 (1994)

    work page 1994

  24. [24]

    M. A. Katsoulakis, N. Mimikos-Stamatopoulos, and B. J. Zhang. Score based generative models are provably robust: an uncertainty quantification perspective. arXiv preprint arXiv:2405.15754 , 2024

    work page arXiv 2024

  25. [25]

    Karras, M

    T. Karras, M. Aittala, T. Aila, and S. Laine. Elucidating the design space of diffusion-based generative models. Advances in neural information processing systems , 35:26565-26577, 2022

    work page 2022

  26. [26]

    Z. Kong, W. Ping, J. Huang, K. Zhao, and B. Catanzaro. Diffwave: A versatile diffusion model for audio synthesis. In International Conference on Learning Representations , 2021

    work page 2021

  27. [27]

    A PDE Perspective on Generative Diffusion Models

    K. Liu, and E. Zuazua. A PDE Perspective on Generative Diffusion Models (2025). ArXiv, abs/2511.05940

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  28. [28]

    X. Li, J. Thickstun, I. Gulrajani, P. S. Liang, and T. B. Hashimoto. Diffusion-lm improves controllable text generation. Advances in Neural Information Processing Systems , 35:4328–4343, 2022

    work page 2022

  29. [29]

    A. Q. Nichol and P. Dhariwal. Improved denoising diffusion probabilistic models. In International conference on machine learning , pages 8162–8171. PMLR, 2021

    work page 2021

  30. [30]

    Poole, A

    B. Poole, A. Jain, J. T. Barron, and B. Mildenhall. Dreamfusion: Text-to-3d using 2d diffusion. In The Eleventh International Conference on Learning Representations , 2023

    work page 2023

  31. [31]

    Popov, I

    V. Popov, I. Vovk, V. Gogoryan, T. Sadekova, and M. Kudinov. Grad-tts: A diffusion probabilistic model for text-tospeech. In Marina Meila and Tong Zhang, editors, Proceedings of the 38th International Conference on Machine Learning, volume 139 of Proceedings of Machine Learning Research, pages 8599–8608. PMLR, 18–24 Jul 2021

    work page 2021

  32. [32]

    Y. Song, S. Ermon. Generative modeling by estimating gradients of the data distribution. Advances in neural information processing systems , 32, 2019

    work page 2019

  33. [33]

    Y. Song, S. Ermon. Improved techniques for training scorebased generative models. Advances in neural in- formation processing systems , 33:12438–12448, 2020. 21

    work page 2020

  34. [34]

    Shawe-Taylor, N

    J. Shawe-Taylor, N. Cristianini, et al., Kernel methods for pattern analysis , Cambridge university press, 2004

    work page 2004

  35. [35]

    Sohl-Dickstein, E

    J. Sohl-Dickstein, E. Weiss, N. Maheswaranathan, and S. Ganguli. Deep unsupervised learning using nonequi- librium thermodynamics. In International conference on machine learning , pages 2256–2265. PMLR, 2015

    work page 2015

  36. [36]

    Y. Song, J. Sohl-Dickstein, D. P. Kingma, A. Kumar, S. Ermon, and B. Poole. Score-based generative modeling through stochastic differential equations. arXiv preprint arXiv:2011.13456 , 2020

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  37. [37]

    Saharia, W

    C. Saharia, W. Chan, H. Chang, C. Lee, J. Ho, T. Salimans, D. Fleet, and M. Norouzi. Palette: Image-toimage diffusion models. In ACM SIGGRAPH 2022 conference proceedings , pages 1–10, 2022

    work page 2022

  38. [38]

    A. Thiery. Denoising diffusion probabilistic models. https://alexxthiery.github.io/notes/DDPM/DDPM.html, 2023

    work page 2023

  39. [39]

    Toscani, Kinetic models of opinion formation, Commun

    G. Toscani, Kinetic models of opinion formation, Commun. Math. Sci., 4 (2006), pp. 481–496

    work page 2006

  40. [40]

    Toscani, M

    G. Toscani, M. Zanella. Supercritical Fokker-Planck equations for consensus dynamics: large-time behaviour and weighted Nash-type inequalities. Ricerche mat 75, 405-429 (2026)

    work page 2026

  41. [41]

    L. Weng. What are diffusion models? https://lilianweng.github. io/posts/2021-07-11-diffusion-models/, Jul 2021. 22

    work page 2021