  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.
