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Physics-based Probing and Prediction of Extreme Events

By Mohammad Farazmand and Themistoklis P. Sapsis

Extreme events arise spontaneously in a variety of natural and engineering systems. They are usually unexpected, transient phenomena that take place over short time scales and have very large magnitudes when compared to typical system responses. In this sense, extreme events are a subclass of the so-called rare events. But while rare events are only associated with low probability, extreme events are also generally of very large magnitude [6]. Examples include extreme weather patterns, aeroacoustic instabilities in combustion engines, earthquakes, rogue waves on the ocean surface, and power grid overloads (see Figure 1). Since these events often have undesirable economic, environmental, and humanitarian consequences, their study is of great interest.

Figure 1. Three examples of extreme events caused by randomly-triggered instabilities of finite lifetime. 1a. Extreme bursts of dissipation in Kolmogorov flow; the vorticity field is shown. 1b. Extreme event in dispersive wave turbulence; the magnitude of the wave field is shown. 1c. A rogue wave in two-dimensional directional seas. The surface shows the wave elevation. Image credit: Mohammad Farazmand and Themistoklis Sapsis.

The four fundamental questions related to extreme events include the following:

  1. Mechanism: What conditions lead to the occurrence of an extreme event?
  2. Quantification: What is the likelihood or frequency of an extreme event taking place?
  3. Prediction: Are there indicators or triggers whose measurement would signal a forthcoming extreme event?
  4. Mitigation: What control strategies are best suited for suppressing extreme events?  Can controlling the trigger lead to suppression? While mitigation of extreme events in nature appears to be out of reach, specialized control protocols may be devised to suppress those in engineered systems (such as power grids or fluid flows).

The first question is of primary focus, since its answer is the cornerstone for the subsequent three questions.

We concentrate on problems for which a model, in the form of a deterministic or stochastic dynamical system, is available. An extreme event is associated with unusual growth (or decay) of the time series of a particular observable of the system. These dramatic fluctuations occur when the system’s trajectory visits a subset of the state space that acts as the basin of attraction to the extreme events. These basins harbor certain instabilities that momentarily repel the trajectory from its background dynamics (see Figure 2). To understand the underpinning mechanism, one must detect the extreme event instability regions of the state space.

Figure 2. Schematic description of extreme events. 2a. Time series of certain observables show intermittent bursts. 2b. Observed bursts correspond to the transient deviation of the system trajectory from the background attractor when it visits extreme event instability regions. Adapted from [5].

In high-dimensional chaotic systems, determining the instability regions is complicated. We propose a constrained optimization method to probe the state space of high-dimensional dynamical systems in search of the subsets that underpin extreme events [5].

Let the governing equations \(\partial_t u = N(u)\) describe a system where \(u(t)\in X\) denotes the state of the system at time \(t\) and \(X\) is an appropriate function space. Also, let \(I:X\to \mathbb R\) denote an observable whose relatively large values constitute an extreme event. In order to probe the onset of extreme events in the state space, we seek initial states \(u_0\in X\) whose corresponding trajectory \(u(t)\) maximizes the growth of the observable \(I\) over a given finite time \(\tau\). To obtain physically relevant maximizers, one must further constrain the initial states \(u_0\) to ensure that they belong to the system’s attractor; this rules out exotic maximizers that have zero probability of being observed under the system’s long-term natural dynamics. This constrained optimization problem can be written more precisely as

\[\sup_{u_0\in X} \left[ I(u(\tau))-I(u(0))\right], \tag{1a}\]

\[\mbox{such that} \begin{cases} \partial_t u = N(u),\quad u(0)=u_0\\
\underline c_i\leq C_i(u_0)\leq \overline c_i,\quad i=1,2,\cdots, k, \tag{1b} \end{cases}\]

where \(\tau\in\mathbb R^+\) is the typical timescale of the extreme events. The constraints involving \(C_i:X\to \mathbb R\) and \(\underline c_i,\overline{c}_i\in \mathbb R\) are enforced to ensure that the maximizer belongs to the attractor (or at least to a small neighborhood of the attractor).

We apply this approach to the Kolmogorov flow, a two-dimensional Navier-Stokes equation driven by sinusoidal shear [5]. The energy dissipation rate is known to exhibit intermittent bursts along the system’s trajectories [3]. Applying an instantaneous version of the optimization problem \((1)\)—and exploiting the energy-conserving nature of the nonlinear term in the Navier-Stokes equation—reveals that the spontaneous transfer of energy to the mean flow from a large-scale Fourier mode causes the dissipation bursts.

Discovery of this mechanism led to an indicator (the energy of the large-scale Fourier mode) whose low values signal an upcoming burst of energy dissipation. Using long-term simulations and Bayesian statistics, we quantify the probability of future extreme events \(P_{ee}\) in terms of the indicator’s current value. This results in short-term prediction of extreme energy dissipation in the Kolmogorov flow (see Figure 3).

Figure 3. Prediction of extreme events in the Kolmogorov flow. 3a. Time series of the energy dissipation rate along a trajectory of the Kolmogorov flow. 3b. Conditional probability density of the future energy dissipation and the indicator. Note that large future dissipation correlates strongly with small present values of the indicator, and vice versa. 3c. Prediction of the extreme event marked with a red box in panel 3a. Pee measure the probability of upcoming extreme events. Adapted from [5].

Time evolution of the plot shown in Figure 3c.

The developed framework is also valuable in the identification of precursors for extreme events in nonlinear water waves, commonly referred to as rogue waves. Our approach employs the wave field’s decomposition into a discrete set of localized wave groups with optimal length scales and amplitudes. These wave groups do not interact due to the prediction’s short-term character; therefore, their dynamics can be characterized individually. Using direct numerical simulations of the governing envelope equations [2], we precomputed the expected maximum elevation for each wave group. The combination of the wave field decomposition algorithm and the precomputed map for expected wave group elevation allows one to (i) understand how the probability of rogue wave occurrence changes as the spectrum parameters vary, (ii) compute a critical length scale characterizing wave groups with high probability of evolving to rogue waves, and (iii) formulate a robust and parsimonious reduced-order prediction scheme for large waves.

Figure 4a displays the contours of the probability density function describing the occurrence of wave groups with length scales in the \(x\)- and \(y\)-direction—\(L_x\) and \(L_y\) respectively—and amplitude \(A_0\). These wave groups are the direct consequence of the dispersive mixing between different wavenumbers. The gray surface marks the parametric boundary above which individual wave groups become unstable. Figure 4b shows the maximal growth occurring over finite time. The parameter space’s low-dimensionality allows us to easily identify a critical pair of length scales that can be tracked to forecast waves with realistic probability of occurrence but also significant growth over finite times. The resulting prediction scheme permits the data-driven prediction of rogue waves occurring due to nonlinear effects without solving any wave equations. This strategy predicts the rogue waves on average approximately 100 wave periods ahead of time [1,4].

Figure 4. Likelihood of dangerous wave groups that lead to rogue waves. 4a. Contours of the probability density function for the occurrence of wave groups with different size. 4b. Contours of maximum finite-time growth of wave groups due to nonlinear focusing phenomena. The combination of statistics and dynamics offers a critical set of length scales relevant to the prediction of extreme wave groups. Adapted from [4].

A random wave field is generated and propagated under the modified nonlinear Schrödinger equation. Wavegroup marked with the red curve is predicted to develop into a rogue wave in the future.

We conclude with the following remarks:

(i) For complex high-dimensional systems, knowledge of the physical model does not imply knowledge of the mechanism underlying extreme events. Constrained optimization \((1)\) offers one systematic method for the discovery of precursors to extreme events by carefully probing the dynamical system’s state space.

(ii) Direct numerical simulations, although insightful, are not adequate for understanding extreme events. Only a low-dimensional subset of the many interacting degrees of freedom in high-dimensional systems contributes to extreme event formation. However, because of the complex coupling among all degrees of freedom, it is unclear how one would implement a data-driven approach to isolate the ones that underpin extreme events. Nevertheless, such a data-driven analysis merits further investigation.

(iii) Knowledge of the mechanism underpinning extreme events may enable the construction of indicators whose measurement permits the data-driven prediction of upcoming extreme events. In other words, upon discovering an indicator of extreme events from the model-based optimization (1), prediction can be accomplished in a completely data-driven fashion without resorting to the model.

Acknowledgments: We acknowledge support from Office of Naval Research grant N00014-15-1-2381, Air Force Office of Scientific Research grant FA9550-16-1-0231, and Army Research Office grant 66710-EG-YIP.

[1] Cousins, W., & Sapsis, T.P. (2016). Reduced-order precursors of rare events in unidirectional nonlinear water waves. J. Fluid Mech, 790, 368-388.
[2] Dysthe, K., Krogstad, H.E., &  Müller, P. (2008). Oceanic rogue waves. Annu. Rev. Fluid Mech., 40, 287-310.
[3] Farazmand, M. (2016). An adjoint-based approach for finding invariant solutions of Navier-Stokes equations. J. Fluid Mech., 795, 278-312.
[4] Farazmand, M., & Sapsis, T.P. (2017). Reduced-order prediction of rogue waves in two-dimensional deep-water waves. J. Comput. Phys., 340, 418-434.
[5] Farazmand, M., & Sapsis, T.P. (2017). A variational approach to probing extreme events in turbulent dynamical systems. Sci. Adv., 3(9), e1701533.
[6] Lucarini, V., Faranda, D., de Freitas, A.C.G.M.M., de Freitas, J.M.M., Holland, M., Kuna, T.,...,Vaienti, S. (2016). Extr. Recur. Dynam. Syst. New York, NY: Wiley.

Mohammad Farazmand is a postdoctoral associate at the Massachusetts Institute of Technology. Themistoklis Sapsis is an associate professor of mechanical engineering at the Massachusetts Institute of Technology.

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