Lyapunov analysis of multiscale dynamics: the slow bundle of the two-scale Lorenz 96 model

Carlu, Mallory; Ginelli, Francesco; Lucarini, Valerio; Politi, Antonio

doi:10.5194/npg-26-73-2019

Cited by 17 publications

(16 citation statements)

References 50 publications

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“…Such techniques have been developed for variational data assimilation (e.g. Trémolet, 2006;Carrassi and Vannitsem, 2010), for ensemble Kalman filters and 144 M. Bocquet et al: Data assimilation as a learning tool to infer dynamics smoothers (e.g. Ruiz et al, 2013;Raanes et al, 2015), and for ensemble-variational assimilation (Amezcua et al, 2017;Sakov et al, 2018).…”

Section: Data Assimilation and Model Errormentioning

confidence: 99%

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Data assimilation as a learning tool to infer ordinary differential equation representations of dynamical models

Bocquet

Brajard

Carrassi

et al. 2019

Nonlin. Processes Geophys.

139

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Abstract. Recent progress in machine learning has shown how to forecast and, to some extent, learn the dynamics of a model from its output, resorting in particular to neural networks and deep learning techniques. We will show how the same goal can be directly achieved using data assimilation techniques without leveraging on machine learning software libraries, with a view to high-dimensional models. The dynamics of a model are learned from its observation and an ordinary differential equation (ODE) representation of this model is inferred using a recursive nonlinear regression. Because the method is embedded in a Bayesian data assimilation framework, it can learn from partial and noisy observations of a state trajectory of the physical model. Moreover, a space-wise local representation of the ODE system is introduced and is key to coping with high-dimensional models. It has recently been suggested that neural network architectures could be interpreted as dynamical systems. Reciprocally, we show that our ODE representations are reminiscent of deep learning architectures. Furthermore, numerical analysis considerations of stability shed light on the assets and limitations of the method. The method is illustrated on several chaotic discrete and continuous models of various dimensions, with or without noisy observations, with the goal of identifying or improving the model dynamics, building a surrogate or reduced model, or producing forecasts solely from observations of the physical model.

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Section: Data Assimilation and Model Errormentioning

confidence: 99%

“…(25) is 0.72, which will be the key timescale when focusing on the slow variables (see e.g. Carlu et al, 2019).…”

mentioning

confidence: 99%

Data assimilation as a learning tool to infer ordinary differential equation representations of dynamical models

Bocquet

Brajard

Carrassi

et al. 2019

Nonlin. Processes Geophys.

139

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“…The other parameters: c = 10 for the time-scale ratio, b = 10 for the space-scale ratio, h = 1 for the coupling, and F = 10 for the forcing, are set to their original values. When uncoupled (h = 0), the Lyapunov time of the slow variables x sector of the model(29)is 0.72, which is the key time scale when focusing on the slow variables[11]. The standard deviation of any of the slow variables is 3.54.The vector u represents unresolved scales and hence model error when only considering the slow variables x.…”

mentioning

confidence: 99%

Bayesian inference of chaotic dynamics by merging data assimilation, machine learning and expectation-maximization

Bocquet¹,

Brajard²,

Carrassi³

et al. 2020

Foundations of Data Science

115

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The reconstruction from observations of high-dimensional chaotic dynamics such as geophysical flows is hampered by (i) the partial and noisy observations that can realistically be obtained, (ii) the need to learn from long time series of data, and (iii) the unstable nature of the dynamics. To achieve such inference from the observations over long time series, it has been suggested to combine data assimilation and machine learning in several ways. We show how to unify these approaches from a Bayesian perspective using expectation-maximization and coordinate descents. In doing so, the model, the state trajectory and model error statistics are estimated all together. Implementations and approximations of these methods are discussed. Finally, we numerically and successfully test the approach on two relevant low-order chaotic models with distinct identifiability.

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“…The other parameters are set to their original values: c = 10 for the time-scale ratio, b = 10 for the space-scale ratio, h = 1 for the coupling between the scales, and F = 10 for the forcing. When uncoupled (h = 0), the dimension of the unstable and neutral subspace of the coarse modes model compartment is N 0 = 13 (for a thorough analysis of this dynamical system, see [19]). The stencil of the surrogate model is chosen to be L = 2.…”

Section: Lorenz-05iiimentioning

confidence: 99%

Online learning of both state and dynamics using ensemble Kalman filters

Bocquet¹,

Farchi²,

Malartic³

2021

FoDS

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<p style='text-indent:20px;'>The reconstruction of the dynamics of an observed physical system as a surrogate model has been brought to the fore by recent advances in machine learning. To deal with partial and noisy observations in that endeavor, machine learning representations of the surrogate model can be used within a Bayesian data assimilation framework. However, these approaches require to consider long time series of observational data, meant to be assimilated all together. This paper investigates the possibility to learn both the dynamics and the state online, i.e. to update their estimates at any time, in particular when new observations are acquired. The estimation is based on the ensemble Kalman filter (EnKF) family of algorithms using a rather simple representation for the surrogate model and state augmentation. We consider the implication of learning dynamics online through (ⅰ) a global EnKF, (ⅰ) a local EnKF and (ⅲ) an iterative EnKF and we discuss in each case issues and algorithmic solutions. We then demonstrate numerically the efficiency and assess the accuracy of these methods using one-dimensional, one-scale and two-scale chaotic Lorenz models.</p>

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Lyapunov analysis of multiscale dynamics: the slow bundle of the two-scale Lorenz 96 model

Cited by 17 publications

References 50 publications

Data assimilation as a learning tool to infer ordinary differential equation representations of dynamical models

Data assimilation as a learning tool to infer ordinary differential equation representations of dynamical models

Bayesian inference of chaotic dynamics by merging data assimilation, machine learning and expectation-maximization

Online learning of both state and dynamics using ensemble Kalman filters

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