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.

翻译：生成扩散模型广泛应用于无监督和自监督机器学习中，旨在通过一组已知样本估计的概率分布生成新样本。该模型在复制密集的现实世界内容（如图像、音乐片段或人类语言）方面已展现出令人瞩目的成果。本研究探讨了其在不可压缩流体流动数值模拟中的应用，着眼于在生成网络支持的概率预测框架中融入不可压缩性等物理约束。为此，我们探索了多种基于分数的条件扩散模型，其中无散度约束通过Leray谱投影算子实现，而自回归条件机制旨在稳定远时间视野下的预测流场快照。所提出的模型在基准湍流问题（即Kolmogorov流）上运行，该问题允许进行相当详细的分析和数值处理，从而简化了用于模拟该流动的数值方法的评估。我们通过开展复杂度递增的数值实验，比较已实现扩散模型的优势与局限性并评估其性能，包括：（i）在与训练数据相同时间视野及相似物理条件下的分布内评估；（ii）对训练中未见过的时间视野进行滚动预测；（iii）对明显不同于训练所见流场的分布外测试。特别地，这些结果展示了扩散模型在偏离训练场景的条件下，重现Kolmogorov湍流主要统计特征的能力。