Lagrangian turbulence lies at the core of numerous applied and fundamental problems related to the physics of dispersion and mixing in engineering, bio-fluids, atmosphere, oceans, and astrophysics. Despite exceptional theoretical, numerical, and experimental efforts conducted over the past thirty years, no existing models are capable of faithfully reproducing statistical and topological properties exhibited by particle trajectories in turbulence. We propose a machine learning approach, based on a state-of-the-art Diffusion Model, to generate single-particle trajectories in three-dimensional turbulence at high Reynolds numbers, thereby bypassing the need for direct numerical simulations or experiments to obtain reliable Lagrangian data. Our model demonstrates the ability to quantitatively reproduce all relevant statistical benchmarks over the entire range of time scales, including the presence of fat tails distribution for the velocity increments, anomalous power law, and enhancement of intermittency around the dissipative scale. The model exhibits good generalizability for extreme events, achieving unprecedented intensity and rarity. This paves the way for producing synthetic high-quality datasets for pre-training various downstream applications of Lagrangian turbulence.

翻译：拉格朗日湍流是工程、生物流体、大气、海洋和天体物理学中与弥散和混合相关的众多应用及基础问题的核心。尽管过去三十年间开展了卓越的理论、数值和实验研究，但现有模型均无法忠实再现湍流中粒子轨迹所呈现的统计与拓扑特性。我们提出一种基于最先进扩散模型的机器学习方法，用以生成高雷诺数三维湍流中的单粒子轨迹，从而绕开为获取可靠拉格朗日数据而进行的直接数值模拟或实验。该模型能够定量复现整个时间尺度范围内的所有相关统计基准，包括速度增量的肥尾分布、异常幂律以及耗散尺度附近间歇性的增强。模型对极端事件展现出良好的泛化能力，达到了前所未有的强度和罕见度。这为生成用于预训练拉格朗日湍流各种下游应用的高质量合成数据集铺平了道路。