近日，荷兰埃因霍温理工大学Xander M. de Wit团队研究了数据驱动的湍流中拉格朗日粒子动力学森-茨万齐格建模。这一研究成果于2026年3月25日发表在《美国科学院院刊》杂志上。

湍流中拉格朗日粒子的动力学在复杂流场的混合、输运和弥散过程中起着关键作用。这些粒子的轨迹展现出高度非平凡的统计行为，这推动了替代模型的发展，旨在无需进行全欧拉场直接数值模拟的高昂计算成本，即可重现这些轨迹。这一任务尤其具有挑战性，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信息。新兴的数据驱动机器学习技术能够在捕捉和重现降阶/替代动力学复杂统计特性方面发挥强大作用。

研究组展示了如何学习一个替代动力系统，使其能够在科尔莫戈罗夫时间尺度上实现短时预测的点态精确性，并在长时间尺度上保持稳定且统计准确，从而演化湍流拉格朗日轨迹。该方法基于森-茨万齐格形式体系，该体系将完整的动力系统进行数学分解：一部分是依赖于当前状态和降阶可观测变量过去历史的可解动力学，另一部分则是由初始状态中未解自由度引起的未解正交动力学。研究组展示了通过基于短时预测的点态误差指标训练该降阶模型，能够正确学习拉格朗日湍流的动力学特性，从而在测试阶段稳定地复现长时间的统计行为。这开辟了一系列应用前景，例如湍流中主动拉格朗日智能体的控制。

附：英文原文

Title: Data-driven Mori–Zwanzig modeling of Lagrangian particle dynamics in turbulent flows

Author: de Wit, Xander M., Gabbana, Alessandro, Woodward, Michael, Lin, Yen Ting, Toschi, Federico, Livescu, Daniel

Issue&Volume: 2026-3-25

Abstract: The dynamics of Lagrangian particles in turbulence play a crucial role in mixing, transport, and dispersion in complex flows. Their trajectories exhibit highly nontrivial 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 applications, for example, for the control of active Lagrangian agents in turbulence.

DOI: 10.1073/pnas.2525390123

Source: https://www.pnas.org/doi/abs/10.1073/pnas.2525390123