The Kuramoto–Sivashinsky equation revisited: Low-dimensional corresponding systems

Khellat, Farhad; Vasegh, Nastaran

doi:10.1016/j.cnsns.2014.01.015

Cited by 4 publications

(3 citation statements)

References 25 publications

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“…= cZ 2 + dZ 3 + Z 5 , Z (10) = aZ 2 + bZ 3 + Z 6 , Z (11) = Z 2 + Z 3 + Z 7 , Z (12) = 2Z 2 + Z 3 + Z 8 , Z (13) = Z 4 + Z 5 + aZ 6 , Z (14) = Z 4 − Z 5 + aZ 6 , Z (15) = Z 1 + aZ 2 + bZ 3 + Z 5 , Z (16) = −Z 1 + aZ 2 + bZ 3 + Z 5 , Z (17) = Z 1 + aZ 2 + bZ 3 + Z 6 , Z (18) = −Z 1 + aZ 2 + bZ 3 + Z 6 , Z (19) = 21) = mZ 2 + mZ 3 + Z 4 + Z 5 , Z (22) = 2aZ 2 + aZ 3 + Z 4 + Z 6 , Z (23) = (1 + a)Z 2 + aZ 3 + Z 4 + Z 7 , Z (24) = (1 + 2a)Z 2 + aZ 3 + Z 4 + Z 8 , Z (25) = aZ 2 + aZ 3 + Z 6 + Z 7 , Z (26) = Z 2 + Z 4 + Z 7 + mZ 8 ,…”

Section: Theorem On Projectionsunclassified

Symmetry reductions of a generalized Kuramoto–Sivashinsky equation via equivalence transformations

Rosa

Bruzón

2018

Communications in Nonlinear Science and Numerical Simulation

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Section: Theorem On Projectionsunclassified

Symmetry reductions of a generalized Kuramoto–Sivashinsky equation via equivalence transformations

Rosa

Bruzón

2018

Communications in Nonlinear Science and Numerical Simulation

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“…Applications of the K-S equation include modeling of the dynamics of self-focusing lasers [51], instabilities in thin films [8], and the flow of a viscous fluid down a vertical plane [57]. Extensive numerical investigations of the chaotic dynamics of the K-S equation have been carried out [20,37,38,44,58]. Furthermore, the K-S equation has been a source of many results related to dynamics of chaotic systems [14,23,53].…”

Section: Kuramoto-sivashinsky Equationmentioning

confidence: 99%

A multiresolution ensemble Kalman filter using the wavelet decomposition

Hickmann

Godinez

2017

Comput Geosci

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We present a method of using classical wavelet based multiresolution analysis to separate scales in model and observations during data assimilation with the ensemble Kalman filter. In many applications, the underlying physics of a phenomena involve the interaction of features at multiple scales. Blending of observational and model error across scales can result in large forecast inaccuracies since large errors at one scale are interpreted as inexact data at all scales. Our method uses a transformation of the observation operator in order to separate the information from different scales of the observations. This naturally induces a transformation of the observation covariance and we put forward several algorithms to efficiently compute the transformed covariance. Another advantage of our multiresolution ensemble Kalman filter is that scales can be weighted independently to adjust each scale's effect on the forecast. To demonstrate feasibility we present applications to a one dimensional Kuramoto-Sivashinsky (K-S) model with scale dependent observation noise and an application involving the forecasting of solar photospheric flux. The latter example demonstrates the multiresolution ensemble Kalman filter's ability to account for scale dependent model error. Modeling of photospheric magnetic flux transport is accomplished by the Air Force Data Assimilative Photospheric Transport (ADAPT) model.

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“…which is a prototypical model of spatiotemporal chaos. Like Khellat and Vasegh [29], we choose the initial condition…”

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confidence: 99%

Influence of database noises to machine learning for spatiotemporal chaos

Yang,

Qin,

Liao

2021

Preprint

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A new strategy, namely the "clean numerical simulation" (CNS), was proposed (J. Computational Physics, 418:109629, 2020) to gain reliable/convergent simulations (with negligible numerical noises) of spatiotemporal chaotic systems in a long enough interval of time, which provide us benchmark solution for comparison. Here we illustrate that machine learning (ML) can always give good enough fitting predictions of a spatiotemporal chaos by using, separately, two quite different training sets: one is the "clean database" given by the CNS with negligible numerical noises, the other is the "polluted database" given by the traditional algorithms in single/double precision with considerably large numerical noises. However, even in statistics, the ML predictions based on the "polluted database" are quite different from those based on the "clean database". It illustrates that the database noises have huge influences on ML predictions of some spatiotemporal chaos, even in statistics. Thus, we must use a "clean" database for machine learning of some spatiotemporal chaos. This surprising result might open a new door and possibility to study machine learning.

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The Kuramoto–Sivashinsky equation revisited: Low-dimensional corresponding systems

Cited by 4 publications

References 25 publications

Symmetry reductions of a generalized Kuramoto–Sivashinsky equation via equivalence transformations

Symmetry reductions of a generalized Kuramoto–Sivashinsky equation via equivalence transformations

A multiresolution ensemble Kalman filter using the wavelet decomposition

Influence of database noises to machine learning for spatiotemporal chaos

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