Divergence-Free Diffusion Models for Incompressible Fluid Flows

2026-01-27

AI Summary

This work addresses the challenge that generative diffusion models often fail to satisfy the divergence-free constraint required for incompressible fluid flows in turbulence forecasting. To this end, the authors propose a score-based conditional diffusion model that, for the first time, integrates Leray spectral projection directly into the generative process to rigorously enforce solenoidal velocity fields. Furthermore, an autoregressive conditioning mechanism is introduced to enhance the stability of long-horizon roll-out predictions. The method is validated on the Kolmogorov turbulence benchmark, demonstrating its ability to accurately reproduce key statistical properties across in-distribution, out-of-distribution, and extended-time simulations. The approach achieves both strong physical consistency—through strict adherence to the incompressibility condition—and robust generalization capabilities.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Machine Learning: Deep Generative Models & Autoencoders
• Natural Language Processing: Generation

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Large language models for search
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

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.

Problem

• incompressible fluid flows
• divergence-free constraint
• generative diffusion models
• physical constraints
• turbulence simulation

Innovation

• divergence-free
• diffusion models
• incompressible flow
• Leray projector
• autoregressive conditioning