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 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 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、海洋及天体物理学中涉及扩散与混合现象的众多应用问题和基础问题的核心。尽管过去三十年间开展了大量理论、数值和实验研究，但现有模型均无法忠实再现湍流粒子轨迹所展现的统计与拓扑特性。本文提出一种基于先进扩散模型的机器学习方法，用于生成高雷诺数三维湍流中的单粒子轨迹，从而绕开直接数值模拟或实验来获取可靠拉格朗日数据的需求。该模型能够再现跨时间尺度的多数统计基准特征，包括速度增量的肥尾分布、反常幂律以及耗散尺度附近间歇性增强现象。在耗散尺度以下观察到轻微偏差，主要体现在加速度和平坦度统计量中。令人惊讶的是，模型对极端事件展现出强泛化能力，能生成强度更高、罕见度更大且仍符合真实统计特征的事件。这为预训练拉格朗日湍流各类下游应用任务时生成合成高质量数据集铺平了道路。