Hierarchy of extreme-event predictability in turbulence revealed by machine learning

Yuxuan Yang; Chenyu Dong; Gianmarco Mengaldo

nlin.CDMar 14, 2026

Hierarchy of extreme-event predictability in turbulence revealed by machine learning

Authors: Yuxuan Yang, Chenyu Dong, Gianmarco Mengaldo

Organizations: National University of Singapore Singapore

Abstract

Extreme-event predictability in turbulence is strongly state dependent, yet event-by-event predictability horizons are difficult to quantify without access to governing equations or costly perturbation ensembles. Here we train an autoregressive conditional diffusion model on direct numerical simulations of the two-dimensional Kolmogorov flow and use a CRPS-based skill score to define an event-wise predictability horizon. Enstrophy extremes exhibit a pronounced hierarchy: forecast skill persists from $\approx 1$ to $> 4$ Lyapunov times across events. Spectral filtering shows that these horizons are controlled predominantly by large-scale structures. Extremes are preceded by intense strain cores organizing quadrupolar vortex packets, whose lifetime sharply separates long- from short-horizon events. These results identify coherent-structure persistence as a governing mechanism for the predictability of turbulence extremes and provide a data-driven route to diagnose predictability limits from observations.

Figures & tables

Figure 1: Predictability evolution of extreme events prior to occurrence. The heatmap shows ${S}_{e}(T)$ against $t-t_{e}$ (bottom axis) and normalized time $(t-t_{e})/T_{\lambda}$ (top axis), with darker shadings indicating higher predictability. Vertical ticks to the left of each shading mark the individual predictability limit (horizons) $T^{*}_{e}$ for each corresponding extreme event. The box plot above the heatmap summarizes the distribution of the event-wise predictability score ( $S_{e}(T)$ ) for all events. The right axis indicates the predictability regime of each event (high predictability, medium predictability, or low predictability) based on its corresponding $T^{*}_{e}$ .

Figure 2: Influence of spatial filtering scale $r_{c}$ on predictability limit $T^{*}_{e}$ . (a) Scatter plots of filtered versus baseline $T^{*}_{e}$ . (b) Evolution of $T^{*}_{e}$ for high (orange) and low (blue) predictability limits events. The vertical dashed area ( $\lambda_{c}/L=0.3$ ) marks the region of predictability breakdown. (c) Visualization of structural degradation in vorticity fields, with the baseline presented on the right and the breakdown scale ( $\lambda_{c}/L=0.35$ ) on the left.

Figure 3: (a) Ensemble-averaged evolution of $\Omega$ (green, left axis) and $|Q_{min}|$ (purple, right axis), with shaded regions denoting one standard deviation. The insets display representative snapshots of the $Q$ -criterion and vorticity at the peak of the extreme event. (b) Temporal evolution of coherent-structure statistics within the analysis window centered on the strain core, defined by the maximum of $|Q_{min}|$ . (c) Cumulative distribution functions (CDFs) of structural lifetimes $\tau$ for high-predictability (orange) and low-predictability (blue) events. The inset shows the bootstrap distribution of the mean lifetime difference $\Delta\tau=\langle\tau\rangle_{\mathrm{high}}-\langle\tau\rangle_{\mathrm{low}}$ , yielding $p=0.017$ .

Chaotic Time-Series Prediction

Chaotic time-series prediction studies how to forecast nonlinear systems whose sensitive dependence on initial conditions limits long-range accuracy.

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Dynamics-Informed Deep Learning for Predicting Extreme Events

Eirini Katsidoniotaki, Themistoklis P. Sapsis

Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139

Chaotic Time-Series Prediction Chaotic Dynamical Systems

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Predicting extreme events in high-dimensional chaotic dynamical systems remains a fundamental challenge, as such events are rare, intermittent, and arise from transient dynamical mechanisms that are difficult to infer from limited observations. Accordingly, real-time forecasting calls for precursors that encode the mechanisms driving extremes, rather than relying solely on statistical associations. We propose a fully data-driven framework for long-lead prediction of extreme events that constructs interpretable, mechanism-aware precursors by explicitly tracking transient instabilities preceding event onset. The approach leverages a reduced-order formulation to compute finite-time Lyapunov exponent (FTLE)-like precursors directly from state snapshots, without requiring knowledge of the governing equations. To avoid the prohibitive computational cost of classical FTLE computation, instability growth is evaluated in an adaptively evolving low-dimensional subspace spanned by Optimal Time-Dependent (OTD) modes, enabling efficient identification of transiently amplifying directions. These precursors are then provided as input to a Transformer-based model, enabling forecast of extreme event observables. We demonstrate the framework on Kolmogorov flow, a canonical model of intermittent turbulence. The results show that explicitly encoding transient instability mechanisms substantially extends practical prediction horizons compared to baseline observable-based approaches.

Abstract

Chaotic Time-Series Prediction

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