Variance Continuity for Lorenz Flows

Bahsoun, Wael; Melbourne, Ian; Ruziboev, Marks

doi:10.1007/s00023-020-00913-5

Cited by 7 publications

(8 citation statements)

References 23 publications

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“…It would be interesting to pursue such an analysis for chaotic ODEs with known mixing rates/decay of correlations. Such results on mixing are less well-developed, however, for chaotic ODEs; see discussion of this point in [68], and the recent work [7].…”

Section: Learning Theory For Markovian Models With Linear Hypothesis ...mentioning

confidence: 99%

“…Note that in both (5.7) and (5.8) 𝜑(⋅) is only evaluated on (compact) 𝒜 obviating the need for any boundedness assumptions on 𝜑(⋅). In the work of Melbourne and co-workers, Assumption A1 is proven to hold for a class of differential equations, including the Lorenz '63 model at, and in a neighbourhood of, the classical parameter values: in [68] the central limit theorem is established; and in [7] the continuity of 𝜎 in 𝜑 is proven. While it is in general very difficult to prove such results for any given chaotic dynamical system, there is strong empirical evidence for such results in many chaotic dynamical systems that arise in practice.…”

Section: Learning Theory For Markovian Models With Linear Hypothesis ...mentioning

confidence: 99%

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A framework for machine learning of model error in dynamical systems

Levine

Stuart

2022

Comm. Amer. Math. Soc.

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The development of data-informed predictive models for dynamical systems is of widespread interest in many disciplines. We present a unifying framework for blending mechanistic and machine-learning approaches to identify dynamical systems from noisily and partially observed data. We compare pure data-driven learning with hybrid models which incorporate imperfect domain knowledge, referring to the discrepancy between an assumed truth model and the imperfect mechanistic model as model error. Our formulation is agnostic to the chosen machine learning model, is presented in both continuous- and discrete-time settings, and is compatible both with model errors that exhibit substantial memory and errors that are memoryless. First, we study memoryless linear (w.r.t. parametric-dependence) model error from a learning theory perspective, defining excess risk and generalization error. For ergodic continuous-time systems, we prove that both excess risk and generalization error are bounded above by terms that diminish with the square-root of T T , the time-interval over which training data is specified. Secondly, we study scenarios that benefit from modeling with memory, proving universal approximation theorems for two classes of continuous-time recurrent neural networks (RNNs): both can learn memory-dependent model error, assuming that it is governed by a finite-dimensional hidden variable and that, together, the observed and hidden variables form a continuous-time Markovian system. In addition, we connect one class of RNNs to reservoir computing, thereby relating learning of memory-dependent error to recent work on supervised learning between Banach spaces using random features. Numerical results are presented (Lorenz ’63, Lorenz ’96 Multiscale systems) to compare purely data-driven and hybrid approaches, finding hybrid methods less datahungry and more parametrically efficient. We also find that, while a continuous-time framing allows for robustness to irregular sampling and desirable domain- interpretability, a discrete-time framing can provide similar or better predictive performance, especially when data are undersampled and the vector field defining the true dynamics cannot be identified. Finally, we demonstrate numerically how data assimilation can be leveraged to learn hidden dynamics from noisy, partially-observed data, and illustrate challenges in representing memory by this approach, and in the training of such models.

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Section: Learning Theory For Markovian Models With Linear Hypothesis ...mentioning

confidence: 99%

Section: Learning Theory For Markovian Models With Linear Hypothesis ...mentioning

confidence: 99%

A framework for machine learning of model error in dynamical systems

Levine

Stuart

2022

Comm. Amer. Math. Soc.

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“…Therefore, the dynamics of the Poincaré map can be understood via quotienting along stable leaves; i.e., by studying the dynamics of its one dimensional quotient map along unstable leaves. The above technique have been employed to obtain statistical properties of Lorenz flows [14] and to prove that such statistical properties of this family of flows is stable under deterministic perturbations [2,7,6]. This illustrates the importance of understanding the statistical properties of one dimensional Lorenz maps.…”

Section: Introductionmentioning

confidence: 98%

Quenched decay of correlations for one dimensional random Lorenz maps

Larkin¹

2021

Preprint

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“…Therefore, the dynamics of the Poincaré map can be understood via quotienting along stable leaves, i.e., by studying the dynamics of its one-dimensional quotient map along stable leaves. The above technique have been employed to obtain statistical properties of Lorenz flows [15] and to prove that such statistical properties of this family of flows is stable under deterministic perturbations [3,7,8]. This illustrates the importance of understanding the statistical properties of one-dimensional Lorenz maps.…”

Section: Introductionmentioning

confidence: 98%

“…Moreover, it admits the so called Sinai-Ruelle-Bowen (SRB) measure, which is ergodic [11]. Its statistical properties, such as mixing rates, limit theorems and their stability under various perturbations, were studied intensively (see for example, [31,10,9,7,14,13,24]).…”

Section: Introductionmentioning

confidence: 99%