Synthetic Lagrangian Turbulence by Generative Diffusion Models*

Lagrangian turbulence is central to numerous applied and fundamental problems concerning the physics of dispersion and mixing across engineering, bio-fluids, atmosphere, oceans, and astrophysics. Despite exceptional theoretical, numerical, and experimental efforts over decades, no current models are capable of accurately reproducing statistical and topological properties of particle trajectories in turbulence. We propose a machine learning approach [1], 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. Surprisingly, 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.