Reduced-order description of transient instabilities and computation of finite-time Lyapunov exponents

Babaee, Hessam; Farazmand, Mohamad; Haller, George; Sapsis, Themistoklis P.

doi:10.1063/1.4984627

Cited by 47 publications

(92 citation statements)

References 42 publications

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“…For hyperbolic fixed points, the OTD subspace aligns with the most unstable eigenspace of the associated linearized operator 32 . For timedependent trajectories, the OTD subspace aligns with the left eigenspace of the Cauchy-Green tensor, which characterizes transient instabilities 34 . As a result, the ith Lyapunov exponent λ i can be recovered from the ith OTD mode:…”

Section: A Preliminariesmentioning

confidence: 99%

Learning the tangent space of dynamical instabilities from data

Blanchard

Sapsis

2019

Chaos: An Interdisciplinary Journal of Nonlinear Science

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For a large class of dynamical systems, the optimally time-dependent (OTD) modes, a set of deformable orthonormal tangent vectors that track directions of instabilities along any trajectory, are known to depend "pointwise" on the state of the system on the attractor, and not on the history of the trajectory. We leverage the power of neural networks to learn this "pointwise" mapping from phase space to OTD space directly from data. The result of the learning process is a cartography of directions associated with strongest instabilities in phase space. Implications for data-driven prediction and control of dynamical instabilities are discussed.The optimally time-dependent (OTD) modes are a set of deformable orthonormal tangent vectors that track directions of instabilities along any trajectory of a dynamical system. Traditionally, these modes are computed by a time-marching approach that involves solving multiple initial-boundary-value problems concurrently with the state equations. However, for a large class of dynamical systems, the OTD modes are known to depend "pointwise" on the state of the system on the attractor, and not on the history of the trajectory. So we propose a neural-network algorithm to learn this "pointwise" mapping from phase space to OTD space directly from data. The neural network produces a cartography of directions of strongest instability in phase space, as well as accurate estimates for the leading Lyapunov exponents.

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Section: A Preliminariesmentioning

confidence: 99%

Learning the tangent space of dynamical instabilities from data

Blanchard

Sapsis

2019

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…One of the recent contributions to the field of slow-fast systems has been the development of accurate numerical methods for computing such invariant manifolds [90,91,92,93]. The computational cost of these man- ifolds increases with the dimension of the system such that their computation is currently limited to four or five dimensional systems [94]. Nonetheless, understanding the mechanism behind extreme events in prototypical low-dimensional slow-fast systems has been greatly informative at the conceptual level.…”

Section: Multiscale Systemsmentioning

confidence: 99%

Extreme Events: Mechanisms and Prediction

Farazmand

Sapsis

2019

Applied Mechanics Reviews

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112

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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.

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“…More precisely, we seek initial states u 0 such that E(u(τ )) − E(u 0 ) is maximized. This is a PDE-constrained optimization problem, since the velocity field u(t) is required to satisfy the channel flow (1).…”

Section: High-likelihood Triggers Of Extreme Eventsmentioning

confidence: 99%

“…Such unstable modes are typically estimated by Dynamic Mode Decomposition (DMD) [38,5,39]. This analysis, however, can only detect modes associated with long-term instabilities which do not seem to explain short-term intermittent events observed in turbulent flows [15,1]. Other variants, however, such as multi-resolution DMD [28] have been demonstrated to work well in systems with relatively low-dimensional attractors.…”

Section: Introductionmentioning

confidence: 99%

Are extreme dissipation events predictable in turbulent fluid flows?

2019

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We derive precursors of extreme dissipation events in a turbulent channel flow. Using a recently developed method that combines dynamics and statistics for the underlying attractor, we extract a characteristic state that precedes laminarization events that subsequently lead to extreme dissipation episodes. Our approach utilizes coarse statistical information for the turbulent attractor, in the form of second order statistics, to identify high-likelihood regions in the state space. We then search within this high probability manifold for the state that leads to the most finite-time growth of the flow kinetic energy. This state has both high probability of occurrence and leads to extreme values of dissipation. We use the alignment between a given turbulent state and this critical state as a precursor for extreme events and demonstrate its favorable properties for prediction of extreme dissipation events. Finally, we analyze the physical relevance of the derived precursor and show its robust character for different Reynolds numbers. Overall, we find that our choice of precursor works well at the Reynolds number it is computed at and at higher Reynolds number flows with similar extreme events.

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Reduced-order description of transient instabilities and computation of finite-time Lyapunov exponents

Cited by 47 publications

References 42 publications

Learning the tangent space of dynamical instabilities from data

Learning the tangent space of dynamical instabilities from data

Extreme Events: Mechanisms and Prediction

Are extreme dissipation events predictable in turbulent fluid flows?

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