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Divergence-Free Diffusion Models for Incompressible Fluid Flows

Wilfried Genuist, Éric Savin, Filippo Gatti, Didier Clouteau

TL;DR

This work investigates divergence-free diffusion models for incompressible fluid flows by enforcing the solenoidal constraint with a Leray projection within an EDM-based diffusion framework and integrating autoregressive conditioning for temporal coherence. The authors generate a 2D Kolmogorov-flow dataset via Fourier-Galerkin discretization and randomized decaying turbulence with Ekman drag, then compare several divergence-free strategies from hard manifold enforcement to soft penalties and corrections. Key findings show that manifold-based divergence-free diffusion yields superior in-distribution accuracy and spectral fidelity, while for long rollouts and out-of-distribution generalization, softer constraints or transport-based corrections offer better robustness. A central design principle emerges: strict, geometry-based constraints excel near training distributions and short horizons, whereas flexible, corrected approaches better handle long-horizon predictions and distributional shifts. The framework is extensible to non-periodic domains and multi-field PDEs and connects diffusion-time dynamics to probability-flow ODE formulations, providing a principled path for physics-informed generative modeling of turbulent flows.

Abstract

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.

Divergence-Free Diffusion Models for Incompressible Fluid Flows

TL;DR

This work investigates divergence-free diffusion models for incompressible fluid flows by enforcing the solenoidal constraint with a Leray projection within an EDM-based diffusion framework and integrating autoregressive conditioning for temporal coherence. The authors generate a 2D Kolmogorov-flow dataset via Fourier-Galerkin discretization and randomized decaying turbulence with Ekman drag, then compare several divergence-free strategies from hard manifold enforcement to soft penalties and corrections. Key findings show that manifold-based divergence-free diffusion yields superior in-distribution accuracy and spectral fidelity, while for long rollouts and out-of-distribution generalization, softer constraints or transport-based corrections offer better robustness. A central design principle emerges: strict, geometry-based constraints excel near training distributions and short horizons, whereas flexible, corrected approaches better handle long-horizon predictions and distributional shifts. The framework is extensible to non-periodic domains and multi-field PDEs and connects diffusion-time dynamics to probability-flow ODE formulations, providing a principled path for physics-informed generative modeling of turbulent flows.

Abstract

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.
Paper Structure (57 sections, 108 equations, 16 figures, 6 tables)

This paper contains 57 sections, 108 equations, 16 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Context and aim of the paper
  3. Related works
  4. Objective and main contributions
  5. Outline of the paper
  6. Incompressible Navier-Stokes equations and dataset generation
  7. Incompressible Navier-Stokes equations
  8. Fourier-Galerkin approximation
  9. Random dataset generation with decaying turbulence
  10. Diffusion models
  11. EDM generative framework
  12. Auto-regressive conditional generation
  13. Divergence-free constrained diffusion models
  14. Divergence-free manifold
  15. Soft divergence-free enforcement
  16. ...and 42 more sections

Figures (16)

  • Figure 1: Snapshots of the vorticity field (rollout up to ${j_\text{max}}=150$).
  • Figure 2: First panel from left to right: empirical density $p_\omega(z;\delta r)$. Second panel: empirical density $p_{\delta u_\parallel}(z;\delta r)$. Third panel: empirical density $p_{\delta\omega_\parallel}(z;\delta r)$. Fourth panel: TKE spectra $k\mapsto E(k)$ from a sampled numerical simulation. The plotted metrics are described in Sect. \ref{['sec:rollout_results']}.
  • Figure 3: In-distribution assessment. Left panel: snapshots of the ground-truth solution (vorticity, top row) against snapshots inferred by vanilla EDM diffusion model ($\mathcal{V}$) (middle row) and by diffusion model on divergence-free manifold (\ref{['eq:EDMdivfree']}) (bottom row). Right panel: zoomed-in regions of vorticity.
  • Figure 4: In-distribution assessment. First panel from left to right: relative $\ell^2$ error ${e_2}$ (lower is better). Second panel: Pearson correlation coefficient $\rho_\text{P}$ (higher is better). Third panel: TKE spectra $k\mapsto E(k)$. Fourth panel: $\varepsilon_\text{div}$, $\operatorname{MSE}$ and $\operatorname{MAE}$ (lower is better). Legends: ($\mathcal{V}$) $\bullet$, (\ref{['eq:EDMdivfree']}) $\blacksquare$, (\ref{['eq:denoiser-Leray']}) $\blacklozenge$, (\ref{['eq:PINN-loss']}) $\blacktriangle$, (\ref{['eq:sample-correction']}) $\blacktriangledown$, (\ref{['eq:PC-Leray']}) ${\mathbf{{\mathbf{x}}}}$. TKE slopes legend: is ground truth spectrum, is $k^{-3-\delta}$ slope.
  • Figure 5: Rollout predictions. Left panel: snapshots of the ground-truth solution (vorticity, top row) against snapshots inferred by vanilla EDM diffusion model ($\mathcal{V}$) (middle row) and by diffusion model on divergence-free manifold (\ref{['eq:EDMdivfree']}) (bottom row). Right panel: zoomed-in regions of vorticity.
  • ...and 11 more figures