mlsanityData to 7 Oct 2026

Divergence-Free Diffusion Models for Incompressible Fluid Flows

arXiv 2601.19368 ↗First posted 27 Jan 2026physics.flu-dyn stat.MLcited by 2

On the map: Diffusion models → Flow matching, time series and science

Abstract (arXiv)

Generative diffusion models are extensively used in unsupervised and self-supervised machine learning with the aim to generate new samples from a probability distribution estimated with a set of known samples. They have demonstrated impressive results in replicating dense, real-world contents such as images, musical pieces, or human languages. This work investigates their application to the numerical simulation of incompressible fluid flows, with a view toward incorporating physical constraints such as incompressibility in the probabilistic forecasting framework enabled by generative networks. For that purpose, we explore different conditional, score-based diffusion models where the divergence-free constraint is imposed by the Leray spectral projector, and autoregressive conditioning is aimed at stabilizing the forecasted flow snapshots at distant time horizons. The proposed models are run on a benchmark turbulence problem, namely a Kolmogorov flow, which allows for a fairly detailed analytical and numerical treatment and thus simplifies the evaluation of the numerical methods used to simulate it. Numerical experiments of increasing complexity are performed in order to compare the advantages and limitations of the diffusion models we have implemented and appraise their performances, including: (i) in-distribution assessment over the same time horizons and for similar physical conditions as the ones seen during training; (ii) rollout predictions over time horizons unseen during training; and (iii) out-of-distribution tests for forecasting flows markedly different from those seen during training. In particular, these results illustrate the ability of diffusion models to reproduce the main statistical characteristics of Kolmogorov turbulence in scenarios departing from the ones they were trained on.

Key facts. Cited by 2 papers in the map (data through 7 Oct 2026). Placed in Flow matching, time series and science, one of 14 directions in Diffusion models. Builds on 59 of its 106 references that are in the map. First posted on arXiv 27 Jan 2026.

Builds on

59 of 106 references.

Closest ancestors · 12

Same direction, newest first

  1. 2025Guiding diffusion models to reconstruct flow fields from sparse data22 Oct
  2. 2025FunDiff: Diffusion Models over Function Spaces for Physics-Informed Generative Modeling9 Jun
  3. 2025Autoregressive regularized score-based diffusion models for multi-scenarios fluid flow prediction30 May
  4. 2024On conditional diffusion models for PDE simulations21 Oct
  5. 2024Integrating Neural Operators with Diffusion Models Improves Spectral Representation in Turbulence Modeling13 Sep
  6. 2024Pi-fusion: Physics-informed diffusion model for learning fluid dynamics6 Jun
  7. 2024PiRD: Physics-informed Residual Diffusion for Flow Field Reconstruction12 Apr
  8. 2023Turbulence Scaling from Deep Learning Diffusion Generative Models10 Nov
  9. 2023Benchmarking Autoregressive Conditional Diffusion Models for Turbulent Flow Simulation4 Sep
  10. 2022A Physics-informed Diffusion Model for High-fidelity Flow Field Reconstruction26 Nov
  11. 2021CSDI: Conditional Score-based Diffusion Models for Probabilistic Time Series Imputation7 Jul
  12. 2021Autoregressive Denoising Diffusion Models for Multivariate Probabilistic Time Series Forecasting28 Jan

Other references · 47

  1. 2017Attention Is All You Needcited by 59,932
  2. 2020Denoising Diffusion Probabilistic Modelscited by 20,851
  3. 2015U-Net: Convolutional Networks for Biomedical Image Segmentationcited by 19,604
  4. 2020Denoising Diffusion Implicit Modelscited by 9,655
  5. 2020Score-Based Generative Modeling through Stochastic Differential Equationscited by 8,964
  6. 2021Diffusion Models Beat GANs on Image Synthesiscited by 8,185
  7. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,005
  8. 2022Classifier-Free Diffusion Guidancecited by 5,717
  9. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,067
  10. 2021Improved Denoising Diffusion Probabilistic Modelscited by 3,736
  11. 2022Elucidating the Design Space of Diffusion-Based Generative Modelscited by 2,994
  12. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
35 more
  1. 2017FiLM: Visual Reasoning with a General Conditioning Layercited by 2,286
  2. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  3. 2023Consistency Modelscited by 1,778
  4. 20163D U-Net: Learning Dense Volumetric Segmentation from Sparse Annotationcited by 1,471
  5. 2022Diffusion Posterior Sampling for General Noisy Inverse Problemscited by 1,375
  6. 2016Group Equivariant Convolutional Networkscited by 1,154
  7. 2021Variational Diffusion Modelscited by 1,145
  8. 2021Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gaugescited by 1,131
  9. 2020Improved Techniques for Training Score-Based Generative Modelscited by 1,036
  10. 2022FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operatorscited by 748
  11. 2019Hamiltonian Neural Networkscited by 702
  12. 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 603
  13. 2021Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modelingcited by 594
  14. 2021Solving Inverse Problems in Medical Imaging with Score-Based Generative Modelscited by 514
  15. 2019Machine Learning for Fluid Mechanicscited by 380
  16. 2021Gotta Go Fast When Generating Data with Score-Based Modelscited by 218
  17. 2022Riemannian Score-Based Generative Modellingcited by 216
  18. 2022Diffusion-based Time Series Imputation and Forecasting with Structured State Space Modelscited by 195
  19. 2022Forecasting Global Weather with Graph Neural Networkscited by 165
  20. 2023The probability flow ODE is provably fastcited by 145
  21. 2021Enhancing Computational Fluid Dynamics with Machine Learningcited by 137
  22. 2024Dynamical Regimes of Diffusion Modelscited by 133
  23. 2022Riemannian Diffusion Modelscited by 120
  24. 2021Soft Truncation: A Universal Training Technique of Score-based Diffusion Model for High Precision Score Estimationcited by 95
  25. 2023Score-based Diffusion Models in Function Spacecited by 80
  26. 2024Provably Robust Score-Based Diffusion Posterior Sampling for Plug-and-Play Image Reconstructioncited by 73
  27. 2022Neural Conservation Laws: A Divergence-Free Perspectivecited by 67
  28. 2023From Zero to Turbulence: Generative Modeling for 3D Flow Simulationcited by 46
  29. 2019Budgeted Training: Rethinking Deep Neural Network Training Under Resource Constraintscited by 36
  30. 2023Manifold Diffusion Fieldscited by 11
  31. 2025Physics-aware generative models for turbulent fluid flows through energy-consistent stochastic interpolantscited by 7
  32. 2025Physics-consistent machine learning: output projection onto physical manifoldscited by 6
  33. 2025Enforcing governing equation constraints in neural PDE solvers via training-free projectionscited by 4
  34. 2024A noise-corrected Langevin algorithm and sampling by half-denoisingcited by 3
  35. 2024Linearly Constrained Diffusion Implicit Modelscited by 1

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2026: 2.

2 papers.

  1. 2026Diffusion Models Observe Only Gradients: A Geometric Perspective on Score Matching Errorscited by 1
  2. 2026Source Anchoring for Physical Consistency in Flow Matching Modelscited by 0

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