mlsanityData to 7 Oct 2026

FP-Diffusion: Improving Score-based Diffusion Models by Enforcing the Underlying Score Fokker-Planck Equation

arXiv 2210.04296 ↗First posted 9 Oct 2022cs.LG cs.AIcited by 36

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

Score-based generative models (SGMs) learn a family of noise-conditional score functions corresponding to the data density perturbed with increasingly large amounts of noise. These perturbed data densities are linked together by the Fokker-Planck equation (FPE), a partial differential equation (PDE) governing the spatial-temporal evolution of a density undergoing a diffusion process. In this work, we derive a corresponding equation called the score FPE that characterizes the noise-conditional scores of the perturbed data densities (i.e., their gradients). Surprisingly, despite the impressive empirical performance, we observe that scores learned through denoising score matching (DSM) fail to fulfill the underlying score FPE, which is an inherent self-consistency property of the ground truth score. We prove that satisfying the score FPE is desirable as it improves the likelihood and the degree of conservativity. Hence, we propose to regularize the DSM objective to enforce satisfaction of the score FPE, and we show the effectiveness of this approach across various datasets.

Key facts. Cited by 36 papers in the map (data through 7 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 20 of its 51 references that are in the map. First posted on arXiv 9 Oct 2022.

Builds on

20 of 51 references.

Closest ancestors · 8

Same direction, newest first

  1. 2022Maximum Likelihood Training for Score-Based Diffusion ODEs by High-Order Denoising Score Matching16 Jun
  2. 2022Score-Based Generative Models Detect Manifolds2 Jun
  3. 2021High-Resolution Image Synthesis with Latent Diffusion Models20 Dec
  4. 2021Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling1 Jun
  5. 2021Diffusion Models Beat GANs on Image Synthesis11 May
  6. 2021Maximum Likelihood Training of Score-Based Diffusion Models22 Jan
  7. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  8. 2020Denoising Diffusion Probabilistic Models19 Jun

Other references · 12

  1. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,005
  2. 2022Photorealistic Text-to-Image Diffusion Models with Deep Language Understandingcited by 6,322
  3. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,067
  4. 2018Neural Ordinary Differential Equationscited by 3,733
  5. 2020DiffWave: A Versatile Diffusion Model for Audio Synthesiscited by 1,339
  6. 2022Denoising Diffusion Restoration Modelscited by 951
  7. 2019Sliced Score Matching: A Scalable Approach to Density and Score Estimationcited by 381
  8. 2017A Downsampled Variant of ImageNet as an Alternative to the CIFAR datasetscited by 374
  9. 2019Flow++: Improving Flow-Based Generative Models with Variational Dequantization and Architecture Designcited by 258
  10. 2021Estimating High Order Gradients of the Data Distribution by Denoisingcited by 63
  11. 2022Self-Consistency of the Fokker-Planck Equationcited by 18
  12. 2020A Modern Analysis of Hutchinson's Trace Estimatorcited by 11

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 5; 2024: 15; 2025: 5; 2026: 11.

36 papers.

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  1. 2023Consistency Trajectory Models: Learning Probability Flow ODE Trajectory of Diffusioncited by 448
  2. 2024Consistent Diffusion Meets Tweedie: Training Exact Ambient Diffusion Models with Noisy Datacited by 44
  3. 2023Characteristic Guidance: Non-linear Correction for Diffusion Model at Large Guidance Scalecited by 26
  4. 2024Training Unbiased Diffusion Models From Biased Datasetcited by 25
  5. 2025Consistent Sampling and Simulation: Molecular Dynamics with Energy-Based Diffusion Modelscited by 23
  6. 2024Diffusion Models as Cartoonists: The Curious Case of High Density Regionscited by 19
  7. 2024Reward-Directed Score-Based Diffusion Models via q-Learningcited by 18
  8. 2025Learning normalized image densities via dual score matchingcited by 17
  9. 2023Exploring the Optimal Choice for Generative Processes in Diffusion Models: Ordinary vs Stochastic Differential Equationscited by 15
  10. 2024Wasserstein proximal operators describe score-based generative models and resolve memorizationcited by 14
  11. 2024Understanding Diffusion Models by Feynman's Path Integralcited by 13
  12. 2023Closing the ODE-SDE gap in score-based diffusion models through the Fokker-Planck equationcited by 12
24 more
  1. 2024Diffusion Rejection Samplingcited by 11
  2. 2026Improving Flow Matching by Aligning Flow Divergencecited by 9
  3. 2024What's the score? Automated Denoising Score Matching for Nonlinear Diffusionscited by 7
  4. 2025Malliavin Calculus for Score-based Diffusion Modelscited by 6
  5. 2024Improving Vector-Quantized Image Modeling with Latent Consistency-Matching Diffusioncited by 5
  6. 2024Bellman Diffusion: Generative Modeling as Learning a Linear Operator in the Distribution Spacecited by 4
  7. 2024HERO: Human-Feedback Efficient Reinforcement Learning for Online Diffusion Model Finetuningcited by 4
  8. 2026RMFlow: Refined Mean Flow by a Noise-Injection Step for Multimodal Generationcited by 3
  9. 2024Neural Shr"odinger Bridge Matching for Pansharpeningcited by 2
  10. 2025Minimal Impact ControlNet: Advancing Multi-ControlNet Integrationcited by 2
  11. 2026Learning Energy-Based Models from Stochastic Interpolants using Spatiotemporal Differencescited by 2
  12. 2023Enhancing Semantic Communication with Deep Generative Models -- An ICASSP Special Session Overviewcited by 1
  13. 2024Efficient Denoising using Score Embedding in Score-based Diffusion Modelscited by 1
  14. 2026Score Shocks: The Burgers Equation Structure of Diffusion Generative Modelscited by 1
  15. 2024Operator-Informed Score Matching for Markov Diffusion Modelscited by 0
  16. 2024Local Curvature Smoothing with Stein's Identity for Efficient Score Matchingcited by 0
  17. 2025Some Optimizers are More Equal: Understanding the Role of Optimizers in Group Fairnesscited by 0
  18. 2026Variance-Reduced Diffusion Sampling via Target Score Identitycited by 0
  19. 2026An Analysis of Regularization and Fokker-Planck Residuals in Diffusion Models for Image Generationcited by 0
  20. 2026Generative Modeling with Flux Matchingcited by 0
  21. 2026Physics-Informed Generative Solver: Bridging Data-Driven Priors and Conservation Laws for Stable Spatiotemporal Field Reconstructioncited by 0
  22. 2026Local Fokker--Planck Geometry for Score Estimation: Heat-Ball Mean-Value Representations and Exact High-Dimensional Samplingcited by 0
  23. 2026Neural Kolmogorov Equations: Parallelizable Learning of Stochastic Dynamics under General Noisecited by 0
  24. 2026Score-based Outlier Generation via Controlling the Radon-Nikodym Derivativecited by 0

Similar, not cited

Close in wording, no citation either way

  1. 2022Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions
  2. 2025A PDE Perspective on Generative Diffusion Models
  3. 2022How Much is Enough? A Study on Diffusion Times in Score-based Generative Models
  4. 2026Diffusion Models Observe Only Gradients: A Geometric Perspective on Score Matching Errors
  5. 2026A Tutorial on Diffusion Theory: From Differential Equations to Diffusion Models
  6. 2024O(d/T) Convergence Theory for Diffusion Probabilistic Models under Minimal Assumptions
  7. 2026A Unified Measure-Theoretic View of Diffusion, Score-Based, and Flow Matching Generative Models
  8. 2023Score-based Diffusion Models in Function Space