Lagrangian turbulence lies at the core of numerous applied and fundamental problems related to the physics of dispersion and mixing in engineering, biofluids, the atmosphere, oceans and astrophysics. Despite exceptional theoretical, numerical and experimental efforts conducted over the past 30 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 reproduce most statistical benchmarks across time scales, including the fat-tail distribution for velocity increments, the anomalous power law and the increased intermittency around the dissipative scale. Slight deviations are observed below the dissipative scale, particularly in the acceleration and flatness statistics. Surprisingly, the model exhibits strong generalizability for extreme events, producing events of higher intensity and rarity that still match the realistic statistics. This paves the way for producing synthetic high-quality datasets for pretraining various downstream applications of Lagrangian turbulence.

拉格朗日湍流是与工程、生物流体、大气、海洋和天体物理学中的分散和混合物理学相关的许多应用和基本问题的核心。尽管过去 30 年在理论、数值和实验方面做出了非凡的努力，但现有的模型还没有能够忠实地再现湍流中粒子轨迹所表现出的统计和拓扑特性。我们提出了一种基于最先进的扩散模型的机器学习方法，可以在高雷诺数的三维湍流中生成单粒子轨迹，从而绕过直接数值模拟或实验来获得可靠的拉格朗日数据。我们的模型展示了跨时间尺度重现大多数统计基准的能力，包括速度增量的肥尾分布、反常幂律和耗散尺度周围增加的间歇性。在耗散标度以下观察到轻微偏差，特别是在加速度和平坦度统计数据中。令人惊讶的是，该模型对极端事件表现出很强的通用性，产生了更高强度和稀有性的事件，但仍然符合实际统计数据。这为生成用于预训练拉格朗日湍流的各种下游应用的合成高质量数据集铺平了道路。