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

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.

Extreme events (EEs) in chaotic dynamics are rare broad excursions whose forecastability can be altered by dynamical noise. We investigate how noise changes EE occurrence and prediction skill across forecast horizons in a third-order autonomous chaotic flow. A single clean-data threshold is frozen for all realizations, broad events are defined by one maximum per excursion, and a future window W=15 is predicted from a 15-time-unit history using HistGradientBoosting with chronological data separation. As the forecast gap G between the observed history and the future event window increases, the clean Matthews correlation coefficient (MCC) decreases from 0.641 at G=0 to 0.165 at G=15. Noise dependence is evaluated with ten paired realizations at eight amplitudes. The mean short-horizon score increases from 0.456 in clean data to 0.546 at sigma=0.007; the paired gain is 0.0895 (95% CI 0.0494-0.1295; Holm-adjusted p=0.0234). Noise strongly increases EE occurrence while event amplitude and width remain comparatively stable. Equalizing positive training counts across noise levels substantially attenuates the short-horizon gain, whereas strong noise reduces intermediate-horizon skill. We term this horizon-dependent, nonuniform change in forecast skill noise-induced predictability redistribution (NIPR).

Deep learning weather models now match numerical weather prediction accuracy while running orders of magnitude faster, but produce deterministic forecasts without uncertainty estimates, a critical gap for high-stakes decisions during extreme weather events. This paper proposes Neural Tangent Kernel-based uncertainty quantification (NTK-UQ) using last-layer empirical features. Theoretical analysis predicts that UQ quality is architecture-dependent through two mechanisms. First, a variance collapse mechanism explains when UQ fails: when the eigenvalue truncation rank approaches the effective rank of the feature space, the GP correction term consumes nearly all prior variance, destroying discrimination between tropical cyclones and routine conditions; architectures with concentrated spectra (spectral operators) require aggressive truncation ($k \leq 10$), while attention-based models tolerate full-rank computation. Second, decomposition performance depends on the non-Gaussian, heavy-tailed structure of extreme weather: Independent Component Analysis exploits higher-order statistics (kurtosis, negentropy) to isolate heavy-tailed extreme-event features, achieving higher discrimination than singular value decomposition, which captures only second-order variance. A data-driven selection rule chooses ICA or SVD from the feature eigenspectrum concentration ratio, correctly prescribing the superior decomposition for all four evaluated architectures. Compared to split conformal prediction (the natural post-hoc baseline), NTK-UQ achieves 31--37% sharper prediction intervals at 90% coverage, and uniquely produces adaptive intervals that scale with extreme event severity, which conformal prediction cannot achieve by construction. The framework requires no retraining; inference-time uncertainty requires only a single matrix-vector product per sample.