The dynamics of Lagrangian particles in turbulence play a crucial role in mixing, transport, and dispersion in complex flows. Their trajectories exhibit highly non-trivial statistical behavior, motivating the development of surrogate models that can reproduce these trajectories without incurring the high computational cost of direct numerical simulations of the full Eulerian field. This task is particularly challenging because reduced-order models typically lack access to the full set of interactions with the underlying turbulent field. Novel data-driven machine learning techniques can be powerful in capturing and reproducing complex statistics of the reduced-order/surrogate dynamics. In this work, we show how one can learn a surrogate dynamical system that is able to evolve a turbulent Lagrangian trajectory in a way that is point-wise accurate for short-time predictions (with respect to Kolmogorov time) and stable and statistically accurate at long times. This approach is based on the Mori-Zwanzig formalism, which prescribes a mathematical decomposition of the full dynamical system into resolved dynamics that depend on the current state and the past history of a reduced set of observables, and the unresolved orthogonal dynamics due to unresolved degrees of freedom of the initial state. We show how by training this reduced order model on a point-wise error metric on short time-prediction, we are able to correctly learn the dynamics of Lagrangian turbulence, such that also the long-time statistical behavior is stably recovered at test time. This opens up a range of new applications, for example, for the control of active Lagrangian agents in turbulence.

翻译：湍流中拉格朗日粒子的动力学在复杂流动的混合、输运与弥散过程中起着关键作用。其轨迹呈现出高度非平凡的统计行为，促使研究者开发能够再现这些轨迹的替代模型，从而避免对完整欧拉场进行直接数值模拟带来的高昂计算成本。这一任务尤为具有挑战性，因为降阶模型通常无法获取与底层湍流场的全部相互作用。新颖的数据驱动机器学习技术能够有效捕捉并再现降阶/替代动力学的复杂统计特性。本研究展示了如何学习一个替代动力系统，使其能够以短时预测（基于柯尔莫哥洛夫时间尺度）点态精确、长时预测稳定且统计精确的方式演化湍流拉格朗日轨迹。该方法基于Mori-Zwanzig形式体系，该体系将完整动力系统数学分解为两部分：依赖于当前状态与降阶观测量历史信息的可解耦动力学，以及由初始状态未解耦自由度导致的不可解正交动力学。我们证明，通过在短时预测上以点态误差度量训练该降阶模型，能够正确学习拉格朗日湍流动力学，使得长时统计行为在测试阶段也能稳定复现。这为湍流中主动拉格朗日智能体的控制等新应用开辟了途径。