Dynamics-Informed Deep Learning for Predicting Extreme Events

TL;DR

This work proposes 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 and demonstrates the framework on Kolmogorov flow, a canonical model of intermittent turbulence.

Abstract

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.

Figures (10)

• Figure 1: An illustration of the first two OTD modes, colored according to their stability properties (blue is stable and red is unstable), along a reference trajectory (green), $\mathbf{u}(t;\mathbf{u}_0)$. A perturbation generates a nearby trajectory (shown in light green color), which undergoes rapid growth along the first OTD direction, resulting in an extreme event.
• Figure 2: Illustration of the training algorithm. A long sequence of snapshots of the system state allows for the approximation of the dynamics (step 2). Dynamics is used to approximate the action of the linearized flow on the OTD subspace, which allows for parsimonious evolution of the OTD modes (step 3). A computation of the FTLE is performed, within the OTD subspace (step 4). Final step 5 is the machine learning of a map from the dominant FTLE to the predicted observable for extreme events.
• Figure 3: Illustration of the prediction steps: 1. Computation of OTD modes; 2. Computation of the associated FTLEs; 3. Prediction of the observable of interest for extreme events.
• Figure 4: Time evolution of the energy dissipation $D(t)$ along a trajectory of Kolmogorov flow with $n = 4$ and $Re = 40$. The signal exhibits small-amplitude background oscillations around $D \approx 0.1$, punctuated by intermittent burst-like excursions corresponding to extreme dissipation events.
• Figure 5: (a) Leading reduced-order finite-time Lyapunov exponent (FTLE), $\hat{\Gamma}_1$, computed within subspaces spanned by $r = 2,\,6,\,8$ OTD modes. The results show that the FTLE converges for $r \ge 6$, whereas smaller subspaces ($r = 2$) underestimate transient growth rates. (b) Leading reduced-order FTLE, $\hat{\Gamma}_1$, computed over different integration horizons $T = 5,\,10,$ and $20\,\mathrm{s}$; short horizons ($T = 5\,\mathrm{s}$) resolve a sequence of localized instability episodes that precede the main event, while increasing $T$ smooths these fluctuations, diminishing the FTLE’s sensitivity as an early-warning indicator. (c) Comparison between data-driven computed FTLE and the one obtained from the analytical variational equations, for $r=6$. (d) Superposition of the dominant FTLE and the observable of interest, $D(t)$ (both quantities are normalized for clarity). The close temporal alignment between the peaks demonstrates that the FTLE effectively captures the buildup of instability preceding dissipation bursts.