Divergence-Free Diffusion Models for Incompressible Fluid Flows

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.

💡 Research Summary

This paper investigates the use of score‑based diffusion models for the probabilistic forecasting of incompressible fluid flows, with a particular focus on enforcing the divergence‑free (incompressibility) constraint in a mathematically rigorous way. Building on the recent “Elucidated Diffusion Model” (EDM) framework, the authors design a diffusion pipeline that (i) employs a stable noise schedule and sampling scheme, (ii) incorporates an autoregressive conditioning mechanism to generate temporally coherent flow sequences, and (iii) guarantees strict incompressibility by applying the Leray spectral projector in Fourier space.

The study is anchored on the two‑dimensional Kolmogorov flow, a canonical benchmark for turbulence that admits a clean Fourier‑Galerkin discretisation and allows precise control of Reynolds number and forcing parameters. Training data consist of many realizations of the flow under varying physical parameters, split into in‑distribution (same time horizon and parameters as seen during training) and out‑of‑distribution (different Reynolds numbers or forcing) test sets.

Two families of models are compared. In the “hard‑constraint” variant, the Leray projection is applied after every denoising step during both training and sampling, thereby removing any compressible component of the velocity field exactly. In the “soft‑constraint” variant, a penalty term proportional to the squared divergence is added to the loss, encouraging but not guaranteeing incompressibility. Both models share the same UNet‑based score network and are trained with denoising score matching.

The autoregressive formulation treats the previous time‑step snapshot as a conditioning input for the diffusion process that predicts the next snapshot. This design enables long‑horizon roll‑outs while mitigating error accumulation, and it is complemented by an optional energy‑corrector that rescales the kinetic energy to the expected level.

Extensive numerical experiments evaluate (a) spectral statistics (energy spectrum, second‑order structure function), (b) information‑theoretic measures (entropy, KL divergence), (c) visual fidelity of vortical structures, and (d) stability of the kinetic energy over long roll‑outs. Results show that the hard‑constraint model reproduces the Kolmogorov‑(k^{-5/3}) spectrum and higher‑order statistics almost indistinguishably from direct numerical simulation (DNS), even for roll‑outs ten times longer than the training horizon. The soft‑constraint model converges faster and uses less memory, but it gradually loses high‑frequency energy in long roll‑outs, leading to a smoother flow field.

Out‑of‑distribution tests demonstrate that both diffusion models generalise better than traditional GAN‑based generators and comparable operator‑learning approaches. The hard‑constraint model, in particular, remains robust when the Reynolds number is increased by a factor of three, preserving statistical signatures of turbulence while delivering inference speeds orders of magnitude faster than DNS.

The authors discuss computational trade‑offs: the Leray projection requires forward and inverse Fourier transforms, adding overhead, whereas the soft penalty is cheap but only approximate. They also note that autoregressive sampling can introduce numerical oscillations in high‑frequency modes, which can be mitigated by the energy‑corrector or by reducing the diffusion step size.

In conclusion, the paper presents a principled, physics‑aware diffusion framework that unifies modern diffusion design, autoregressive temporal modeling, and exact incompressibility enforcement. It establishes that such models can faithfully reconstruct turbulent statistics, generate long‑term coherent flow sequences, and generalise to unseen physical regimes. Future work is outlined toward three‑dimensional Navier‑Stokes, complex geometries, hybrid loss designs combining hard and soft constraints, and integration with control or data‑assimilation pipelines.