Source-linked AI summary
Extreme Events: Mechanisms and Prediction
Mohammad Farazmand, Themistoklis P. Sapsis
TL;DR
Extreme events carry adverse consequences, while chaotic dynamics and uncertainty complicate their prediction. This review examines formation mechanisms and real-time prediction, including mechanism-informed indicators and a variational approach. It describes how a turbulent-flow event involves energy transfer among Fourier modes, and notes that optimizing reliable indicators remains unresolved in the reviewed treatment.
Problem
Extreme events can cause humanitarian, environmental, and financial harm, while chaotic-system sensitivity to initial conditions makes prediction uncertain even when the model is deterministic.
Source: §1 · pp. 1–2Method
The review surveys extreme-event mechanisms and real-time prediction, including a variational mechanism-discovery method and indicators informed by partial mechanism knowledge.
Source: §1 · p. 3, §6 · p. 19Results
In the reviewed Kolmogorov flow, mode a(1,0) loses energy to a(0,k_f) through a triad interaction during extreme events, contributing to bursts in energy input rate.
Source: §4.2 · p. 12Takeaways & Limitations
Mechanism-informed indicators can support real-time prediction, which can be data-driven if the derived indicator is measurable or observable in practice.
Source: §6 · p. 19Optimizing the reliable-indicator objective is deferred because its failure-rate dependence on the indicator is nonlinear and nonsmooth.
Source: §5.3 · p. 18
Abstract
from arXiv · showhide
Extreme events, such as rogue waves, earthquakes and stock market crashes, occur spontaneously in many dynamical systems. Because of their usually adverse consequences, quantification, prediction and mitigation of extreme events are highly desirable. Here, we review several aspects of extreme events in phenomena described by high-dimensional, chaotic dynamical systems. We specially focus on two pressing aspects of the problem: (i) Mechanisms underlying the formation of extreme events and (ii) Real-time prediction of extreme events. For each aspect, we explore methods relying on models, data or both. We discuss the strengths and limitations of each approach as well as possible future research directions.
1 Introduction
Extreme events are difficult to predict in high-dimensional chaotic systems, where early warning signs may be absent, models may be unavailable or poorly informed by sparse extreme-event data, and predictions carry uncertainty. This review centers on mechanisms and real-time prediction, while placing these alongside mitigation and statistics.
1 Introduction
- Rare or absent warning signs, unavailable models, and sparse observations of extremes make direct prediction and data assimilation difficult.In chaotic systems, sensitivity to initial conditions also makes prediction uncertain, even with deterministic models.Source: §1 · pp. 1–2
- Real-time prediction requires measurable observables with early warning signs, and chaotic-system uncertainty means predictions need probabilistic confidence estimates.Reliable indicators should have low false-positive and false-negative rates.Source: §1 · p. 2
- Dynamical-systems mitigation research remains limited to simplified models with arbitrary nudges; systematic control-theoretic study of controllability and observability is missing.The review notes that mitigation may be impractical in natural systems but could be designed for some engineered systems, such as power grids.Source: §1 · p. 2
- Extreme-value theory studies limiting distributions of maxima, while large-deviation theory addresses tail distributions; the review otherwise limits its statistics discussion to this section.The cited review notes that practical extreme-value analysis has extended beyond the independent-and-identically-distributed assumption.Source: §1 · pp. 2–3
- Mechanism knowledge can inform data-driven indicators, control-strategy choices and timing, and estimates of event likelihood and frequency.The review notes that partial mechanism knowledge may help discover reliable indicators.Source: §1 · pp. 2–3
- The review centers on mechanisms, surveying formation mechanisms and a variational discovery framework before discussing reliable indicators for real-time prediction.It also considers model-, data-, and combined approaches to prediction and mitigation, while limiting its statistical discussion to the relevant section.Source: §1 · pp. 2–3
2 Setup and notation
The setup treats a dynamical system's state evolution as generating observable time series, with extreme events defined through threshold exceedances. It also distinguishes such extremes from rare transitions between long-lived states.
2 Setup and notation
- An invariant probability measure on the state space makes the observable along the solution map a continuous stochastic process.The initial state is drawn compatibly with that measure.Source: §2 · p. 3
- An extreme event is defined for an observable using a prescribed threshold; trajectory passage through its event set appears as a burst in the observable time series.The observable may represent quantities such as wave height, temperature, or precipitation.Source: §2 · p. 3, Figure 2 · p. 4
- Rare transitions between long-lived states are not necessarily extreme events under the threshold-based definition, and may be driven by noise.The system can remain near one state for a long time before moving to another and later transitioning back.Source: §2 · p. 3, Figure 3 · p. 4
- The setup covers systems described by partial differential equations and ordinary differential equations, including ODEs obtained by finite-dimensional PDE approximation.The state evolves from an initial condition under a physics-supplied, potentially nonlinear operator and suitable boundary conditions.Source: §2 · pp. 3–4
3 Routes to extreme events
This section reviews mechanisms for extreme events in slow-fast systems, homoclinic and heteroclinic bursting, and noise-induced transitions.
3.1 Multiscale systems
- Slow-fast bursts arise when trajectories are repelled from a normally repelling manifold segment and then return toward an attracting segment; complex repelling sets can make bursts sporadic.With sufficiently strong normal repulsion, excursions away from the slow manifold appear as rapid bursts.Source: §3.1 · pp. 4–5
- For small ε, the critical manifold persists as a nearby slow manifold whose attracting and repelling parts sustain bursting; the illustrated parameter set has a stable periodic orbit.The orbit repeatedly leaves the repelling segment, follows an attracting segment to a fold, and returns toward the other attracting segment.Source: §3.1 · pp. 5–6, Figure 6 · p. 6
- In coupled FitzHugh–Nagumo units, the mean x oscillates chaotically around zero with occasional large bursts, which recur without a regular pattern.Similar extreme events are reported for larger numbers of units and varied parameter values.Source: §3.1 · p. 6, Figure 7 · p. 7
- Computing slow manifolds and their stable and unstable manifolds becomes prohibitive as dimension grows; such computations are currently limited to four- or five-dimensional systems.The mechanisms in low-dimensional slow-fast systems have nevertheless been conceptually informative.Source: §3.1 · p. 7
3.2 Homoclinic and heteroclinic bursting
- A Shilnikov homoclinic orbit can generate infinitely many unstable periodic orbits whose spiraling and unstable-direction repulsion produce small oscillations and occasional bursts; heteroclinic bursting connects different fixed points.Generic trajectories shadowing the unstable periodic orbits can be chaotic.Source: §3.2 · p. 7
3.3 Noise-induced transitions
- For noise-driven transitions, transition-path theory uses committors and transition densities to construct a probability current whose streamlines reveal likely routes between states.The current’s surface flux gives the probability flux of transition trajectories, and the illustrated streamlines show the most probable path.Source: §3.3 · pp. 8–9
- Transition-path theory requires solving three PDEs, making higher-dimensional applications costly and limiting applications to two- and three-dimensional systems.Numerical approximations have been developed to partially address the computational cost.Source: §3.3 · p. 9
4 Variational method for physics-based probing of extreme events
The variational method combines governing equations with information about the system attractor to find states that develop into extreme events. The section illustrates it with turbulent energy-dissipation bursts and oceanic rogue-wave prediction.
4.1 The variational method
- The optimization constrains initial states to an approximate attractor, excluding dynamically irrelevant states and reducing the search to physically relevant precursors.It seeks states that maximize observable growth over a prescribed time interval.Source: §4.1 · pp. 9–10
4.2 Application to a turbulent fluid flow
- At Re=40, triad transfer from a(1,0) to a(0,kf) raises energy input I and coincides with energy-dissipation D bursts; the optimal modes identify this interaction.Because the higher-wavenumber mode has a larger dissipation prefactor, transferring energy to it increases D.Source: §4.2 · pp. 11–13
4.3 Application to oceanic rogue waves
- Localized-wave-group PDFs identify likely future rogue-wave precursors, enabling reduced-order prediction without NLS integration; later work extends it to two-dimensional waves [175].The PDF’s boundary separates groups that do and do not develop into rogue waves.Source: §4.3 · p. 14, Figure 16 · p. 14
5 Prediction of extreme events
The section evaluates indicators by how well they predict future extremes, then illustrates their use in Kolmogorov flow and rogue waves. It also frames indicator discovery as an optimization problem and sketches desirable properties for data-driven methods.
5.1 Conditional statistics for extreme events
- Conditional statistics assess whether an indicator’s threshold separates upcoming extremes from non-extremes with few false positives and false negatives.Pee(g0) gives the probability that the future maximum exceeds the extreme-event threshold at a present indicator value g0.Source: §5.1 · p. 16, Figure 18 · p. 16
5.2 Applications
- In Kolmogorov flow, g = −|a(1,0)| gives Pee ≃ 0 below −0.5, Pee ≃ 1 above −0.3, and a Pee = 0.5 threshold near −0.39.Along one trajectory, Pee rises past 0.5 around t = 30, signaling an imminent event.Source: Figure 19 · p. 17, §5.2 · p. 17
- For rogue waves, small indicator values still leave 0.05 < Pee < 0.2, while false negatives comprise 5.9% of predictions.The conditional density shows more false negatives than for Kolmogorov flow; a more reliable indicator would reduce both false negatives and false positives.Source: §5.2 · p. 17, Figure 20 · p. 18
5.3 Data-driven discovery of indicators
- A systematic framework for discovering indicators directly from data is missing; the proposed criterion is to minimize combined false positives and false negatives.Data-only discovery could be useful where governing equations are unknown, but technical problems remain unresolved.Source: §5.3 · p. 18
- The failure rate combines false-positive and false-negative probabilities, and its minimization must jointly select the indicator, alarm threshold, and prediction horizon.For the Kolmogorov-flow indicator, predictive skill deteriorates as the horizon increases and vanishes for t1 > 4; the nonlinear, nonsmooth optimization remains unresolved.Source: §5.3 · p. 18
6 Summary and conclusions
The review covers mechanisms and real-time prediction, while noting open challenges in data-driven discovery, mitigation, and non-autonomous systems.
6 Summary and conclusions
- Extreme-event mechanisms vary across systems, but the review suggests a possible unified mathematical framework for discovering them.Examples include multiscale instabilities, noise, and nonlinear energy transfers.Source: §6 · p. 19
- For high-dimensional chaotic systems, the review favors combining governing equations with observations or low-order statistics to discover mechanisms.Its variational method uses equations to form an objective functional and data to form constraints.Source: §6 · p. 19
- Reliable indicators can be informed by partial mechanism knowledge, while prediction can be data-driven if the indicator is measurable in practice.Reliability means relatively low false-positive and false-negative rates.Source: §6 · p. 19
- Data-driven indicator discovery remains unresolved, and mitigation research is narrow, with existing studies centered on arbitrary perturbations.The review calls for control-theoretic mitigation framed around measurable observables and adjustable control variables.Source: §6 · p. 19
- The review is limited to autonomous systems; extreme-event research in non-autonomous systems, where rules can change over time, remains sparse.Stock markets and social networks are named as examples of non-autonomous systems.Source: §6 · p. 19