Learning to Forecast Dynamical Systems from Streaming Data

Giannakis, Dimitrios; Henriksen, Amelia; Tropp, Joel A.; Ward, Rachel

doi:10.48550/arxiv.2109.09703

Cited by 4 publications

(4 citation statements)

References 60 publications

(91 reference statements)

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“…An interesting avenue of research would be to study how to optimally recompute the dimensionality reduction step so that it requires the least amount of computational effort. Possible sources of ideas could come from landmarking as in [54] or [55], as well as from streaming PCA algorithms [56], or from [57]. The potential advantage of calculating diffusion map coordinates and working in the reduced space is that (a) the evaluation of the surrogate is cheaper in the reduced space, but also (b) the number of coordinates in which this surrogate is optimized is lower.…”

Section: Discussion and Outlookmentioning

confidence: 99%

On the Correspondence between Gaussian Processes and Geometric Harmonics

Dietrich,

Bello-Rivas,

Kevrekidis

2021

Preprint

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We discuss the correspondence between Gaussian process regression and Geometric Harmonics, two similar kernel-based methods that are typically used in different contexts. Research communities surrounding the two concepts often pursue different goals. Results from both camps can be successfully combined, providing alternative interpretations of uncertainty in terms of error estimation, or leading towards accelerated Bayesian Optimization due to dimensionality reduction.

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Section: Discussion and Outlookmentioning

confidence: 99%

On the Correspondence between Gaussian Processes and Geometric Harmonics

Dietrich,

Bello-Rivas,

Kevrekidis

2021

Preprint

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“…However, a number of recent ideas for scaling kernel methods can be applied, see e.g. [17,33] and references therein. Perhaps more importantly, specific to the context of dynamical systems is the fact that an approximate mode decomposition needs to be further computed, which requires the spectral decomposition of G. As showed in App.…”

Section: Principal Component Regression (Pcr) a Standard Strategy To ...mentioning

confidence: 99%

Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spaces

Kostić¹,

Novelli²,

Maurer³

et al. 2022

Preprint

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We study a class of dynamical systems modelled as Markov chains that admit an invariant distribution via the corresponding transfer, or Koopman, operator. While data-driven algorithms to reconstruct such operators are well known, their relationship with statistical learning is largely unexplored. We formalize a framework to learn the Koopman operator from finite data trajectories of the dynamical system. We consider the restriction of this operator to a reproducing kernel Hilbert space and introduce a notion of risk, from which different estimators naturally arise. We link the risk with the estimation of the spectral decomposition of the Koopman operator. These observations motivate a reduced-rank operator regression (RRR) estimator. We derive learning bounds for the proposed estimator, holding both in i.i.d. and non i.i.d. settings, the latter in terms of mixing coefficients. Our results suggest RRR might be beneficial over other widely used estimators as confirmed in numerical experiments both for forecasting and mode decomposition.

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“…See also measure-preserving EDMD [32]. Other data-driven methods include deep learning [84,86,88,98,141], reduced-order modeling [12,58], sparse identification of nonlinear dynamics [20,108], and kernel analog forecasting [25,55,143]. However, remaining challenges include the following: C1.…”

Section: Introductionmentioning

confidence: 99%

Rigorous data‐driven computation of spectral properties of Koopman operators for dynamical systems

Colbrook

Townsend

2023

Comm Pure Appl Math

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Koopman operators are infinite‐dimensional operators that globally linearize nonlinear dynamical systems, making their spectral information valuable for understanding dynamics. However, Koopman operators can have continuous spectra and infinite‐dimensional invariant subspaces, making computing their spectral information a considerable challenge. This paper describes data‐driven algorithms with rigorous convergence guarantees for computing spectral information of Koopman operators from trajectory data. We introduce residual dynamic mode decomposition (ResDMD), which provides the first scheme for computing the spectra and pseudospectra of general Koopman operators from snapshot data without spectral pollution. Using the resolvent operator and ResDMD, we compute smoothed approximations of spectral measures associated with general measure‐preserving dynamical systems. We prove explicit convergence theorems for our algorithms (including for general systems that are not measure‐preserving), which can achieve high‐order convergence even for chaotic systems when computing the density of the continuous spectrum and the discrete spectrum. Since our algorithms have error control, ResDMD allows aposteri verification of spectral quantities, Koopman mode decompositions, and learned dictionaries. We demonstrate our algorithms on the tent map, circle rotations, Gauss iterated map, nonlinear pendulum, double pendulum, and Lorenz system. Finally, we provide kernelized variants of our algorithms for dynamical systems with a high‐dimensional state space. This allows us to compute the spectral measure associated with the dynamics of a protein molecule with a 20,046‐dimensional state space and compute nonlinear Koopman modes with error bounds for turbulent flow past aerofoils with Reynolds number >105 that has a 295,122‐dimensional state space.

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Learning to Forecast Dynamical Systems from Streaming Data

Cited by 4 publications

References 60 publications

On the Correspondence between Gaussian Processes and Geometric Harmonics

On the Correspondence between Gaussian Processes and Geometric Harmonics

Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spaces

Rigorous data‐driven computation of spectral properties of Koopman operators for dynamical systems

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