Reproducing kernel Hilbert space compactification of unitary evolution groups

Das, Suddhasattwa; Giannakis, Dimitrios; Sławińska, Joanna

doi:10.1016/j.acha.2021.02.004

Cited by 50 publications

(67 citation statements)

References 73 publications

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“…In the former case, if Φ t is governed by a vector field then natural “diffused” versions G ϵ and V ϵ of G and V are provided by normalized forward and backward Kolmogorov equations, respectively. In the latter case, one may apply various numerical schemes 26 , 30 , 34 , 36 , 64 . The scheme 34 is outlined in the Methods section.…”

Section: Resultsmentioning

confidence: 99%

Spectral analysis of climate dynamics with operator-theoretic approaches

et al. 2021

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The Earth’s climate system is a classical example of a multiscale, multiphysics dynamical system with an extremely large number of active degrees of freedom, exhibiting variability on scales ranging from micrometers and seconds in cloud microphysics, to thousands of kilometers and centuries in ocean dynamics. Yet, despite this dynamical complexity, climate dynamics is known to exhibit coherent modes of variability. A primary example is the El Niño Southern Oscillation (ENSO), the dominant mode of interannual (3–5 yr) variability in the climate system. The objective and robust characterization of this and other important phenomena presents a long-standing challenge in Earth system science, the resolution of which would lead to improved scientific understanding and prediction of climate dynamics, as well as assessment of their impacts on human and natural systems. Here, we show that the spectral theory of dynamical systems, combined with techniques from data science, provides an effective means for extracting coherent modes of climate variability from high-dimensional model and observational data, requiring no frequency prefiltering, but recovering multiple timescales and their interactions. Lifecycle composites of ENSO are shown to improve upon results from conventional indices in terms of dynamical consistency and physical interpretability. In addition, the role of combination modes between ENSO and the annual cycle in ENSO diversity is elucidated.

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Section: Resultsmentioning

confidence: 99%

Spectral analysis of climate dynamics with operator-theoretic approaches

et al. 2021

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“…Examples of reduced modeling techniques are linear inverse models [66], (extended) DMD [70,73,84], and methods for approximating the Koopman generator [28,19,46]. These methods formally assume that the training data have a (deterministic) Markovian evolution.…”

Section: Forecasting Methodologiesmentioning

confidence: 99%

Learning to Forecast Dynamical Systems from Streaming Data

Giannakis¹,

Henriksen²,

Tropp³

et al. 2021

Preprint

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Kernel analog forecasting (KAF) is a powerful methodology for data-driven, non-parametric forecasting of dynamically generated time series data. This approach has a rigorous foundation in Koopman operator theory and it produces good forecasts in practice, but it suffers from the heavy computational costs common to kernel methods. This paper proposes a streaming algorithm for KAF that only requires a single pass over the training data. This algorithm dramatically reduces the costs of training and prediction without sacrificing forecasting skill. Computational experiments demonstrate that the streaming KAF method can successfully forecast several classes of dynamical systems (periodic, quasi-periodic, and chaotic) in both data-scarce and data-rich regimes. The overall methodology may have wider interest as a new template for streaming kernel regression.

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“…We now outline an RKHS-based approach [DGS18], which identifies observables satisfying this condition through eigenfunctions of a regularized operator̃on 2 ( ) approximating with the properties of (i) being skew-adjoint and compact; and (ii) having eigenfunctions in the domain of the Nyström operator, which maps them to differentiable functions in an RKHS. Here, is a positive regularization parameter such that, as → 0 + ,̃converges to in a suitable spectral sense.…”

Section: Coherent Pattern Extractionmentioning

confidence: 99%

“…In [DGS18], a constructive procedure was proposed for obtaining the kernel family through a Markov semigroup on 2 ( ). This method has a data-driven implementation, with analogous spectral convergence results for the associated integral operators , on 2 ( ) to those described in the setting of forecasting.…”

Section: Coherent Pattern Extractionmentioning

confidence: 99%

Bridging Data Science and Dynamical Systems Theory

Berry

Giannakis

Harlim

2020

Notices Amer. Math. Soc.

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Modern science is undergoing what might arguably be called a "data revolution," manifested by a rapid growth of observed and simulated data from complex systems, as well as vigorous research on mathematical and computational frameworks for data analysis. In many scientific branches, these efforts have led to the creation of statistical models of complex systems that match or exceed the skill of first-principles models. Yet, despite these successes, statistical models are oftentimes treated as black boxes, providing limited guarantees about stability and convergence as the amount of training data increases. Black-box models also offer limited insights about the operating mechanisms (physics), the understanding of which is central to the advancement of science.

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Reproducing kernel Hilbert space compactification of unitary evolution groups

Cited by 50 publications

References 73 publications

Spectral analysis of climate dynamics with operator-theoretic approaches

Spectral analysis of climate dynamics with operator-theoretic approaches

Learning to Forecast Dynamical Systems from Streaming Data

Bridging Data Science and Dynamical Systems Theory

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