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

Abstract

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.

Figures (5)

• Figure 1: (a) Schematic depiction of the learning task for the Mori-Zwanzig (MZ) surrogate model: feeding a limited ground-truth history, the MZ model aims to predict the continuation of the trajectory in a way that is point-wise accurate at short times (w.r.t. Kolmogorov time $\tau_\eta$) and statistically accurate at longer times. (b) Sketch of the data-driven MZ approach: we begin with the true evolution of the observables $\bm g(t)$ described by the generalized Langevin equation (GLE, \ref{['eq:GLE']}). Applying the projection operator $\mathcal{P}$ eliminates the orthogonal‑dynamics contribution, so the resulting GLE is formulated solely in the projected subspace $\hat{\bm{g}} := \mathcal{P}\bm g$. (c) Exemplary 3D Lagrangian trajectories, comparing ground-truth data (blue) with its corresponding MZ prediction (red). Trajectories are colored by the local vorticity magnitude (darker color means larger vorticity). One can appreciate the development of intermittent vortex-filament structures in certain trajectories (insets). (d) PDFs of Lagrangian velocity differences, normalized by their standard deviation, for different time lags $\tau$. PDFs are vertically shifted for readability.
• Figure 2: Convergence of the accuracy of the predicted PDF of particle acceleration with increasing number of trajectories in the training set from 50K to 5M. Dashed line indicates a Gaussian distribution with the same variance.
• Figure 3: Summary statistics of the long time prediction of the MZ model, compared with the ground truth. Provided are the acceleration PDF (a), its autocorrelation (b), the Q-R diagram of the velocity gradient at the position of the particle (c,d) and the joint PDF of the magnitude of acceleration and velocity gradient (e,f). The dashed line in (a) indicates a Gaussian distribution with the same variance. The lines in (c-f) correspond to level contours of 70%, 90% and 95% probability percentiles.
• Figure 4: Temporal robustness of the MZ model. (a) the mean-squared error (MSE) of acceleration, normalized by twice the mean square (MS) acceleration, as a function of time. (b) PDFs of acceleration computed over different time windows. The dashed line indicates a Gaussian distribution with the same variance.
• Figure 5: Model predictions when starting from statistically perturbed zero initial conditions. (a) KL-divergence $D_{\textrm{KL}}(\textrm{ground-truth}\, || \,\textrm{model})$ based on the distribution of particle acceleration as a function of time throughout the predicted trajectories. (b) corresponding PDFs of acceleration for selected time windows. The dashed line indicates a Gaussian distribution with the same variance. (c) Q-R diagram of the predicted velocity gradient at the particle position, averaged from $t>20\tau_\eta$.