Modeling of missing dynamical systems: deriving parametric models using a nonparametric framework

Jiang, Shixiao W.; Harlim, John

doi:10.1007/s40687-020-00217-4

Cited by 19 publications

(27 citation statements)

References 42 publications

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“…In view of statistical learning of the high-dimensional nonlinear flow map in (2.6), each linear-in-parameter reduced model provides an optimal approximation to the flow map in the function space spanned by the proposed terms. A possible future direction is to select adaptive-to-data hypothesis spaces in a nonparametric fashion [23] and analyze the distance between the flow map and the hypothesis space spanned by these proposed terms [45,46].…”

Section: Model Selectionmentioning

confidence: 99%

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Data-driven Model Reduction for Stochastic Burgers Equations

2020

Preprint

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We present a class of efficient parametric closure models for 1D stochastic Burgers equations. Casting it as statistical learning of the flow map, we derive the parametric form by representing the unresolved high wavenumber Fourier modes as functionals of the resolved variables’ trajectory. The reduced models are nonlinear autoregression (NAR) time series models, with coefficients estimated from data by least squares. The NAR models can accurately reproduce the energy spectrum, the invariant densities, and the autocorrelations. Taking advantage of the simplicity of the NAR models, we investigate maximal and optimal space-time reduction. Reduction in space dimension is unlimited, and NAR models with two Fourier modes can perform well. The NAR model’s stability limits time reduction, with a maximal time step smaller than that of the K-mode Galerkin system. We report a potential criterion for optimal space-time reduction: the NAR models achieve minimal relative error in the energy spectrum at the time step where the K-mode Galerkin system’s mean CFL number agrees with the full model’s.

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Section: Model Selectionmentioning

confidence: 99%

“…These developments demand a systematic understanding of model reduction from the perspectives of dynamical systems (see, e.g. [7,19,20]), numerical approximation [21,22], and statistical learning [23].…”

Section: Introductionmentioning

confidence: 99%

Data-driven Model Reduction for Stochastic Burgers Equations

2020

Preprint

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“…Constructing RKHS via orthogonal polynomials. Inspired by the Mercer's theorem [48] and reproducing kernel weighted Hilbert space used in [5,22,23,58], we consider the orthonormal polynomials with respect to L 2 (R d , W ) to construct our kernel and RKHS. Here,…”

Section: Proof To Begin With By Lemma 23k Defines a Kernel Andk(•mentioning

confidence: 99%

“…Indeed, for data that lie on a smooth d-dimensional manifold embedded in R n , one can construct a Mercer-type kernel based on the orthogonal basis functions obtained via the diffusion maps algorithm [9] and achieve the optimal rate as reported in [49]. While this approach has been explored in [23], the errors in the estimation of these eigenbasis,…”

mentioning

confidence: 99%

Estimating linear response statistics using orthogonal polynomials: An rkhs formulation

Zhang¹,

Harlim²,

Li³

2020

Foundations of Data Science

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“…The past decades witness revolutionary developments of data-driven strategies, ranging from parametric models (see, e.g., [ 8 , 9 , 10 , 11 , 12 , 13 , 14 ] and the references therein) to nonparametric and machine learning methods (see, e.g., [ 15 , 16 , 17 , 18 ]). These developments demand a systematic understanding of model reduction from the perspectives of dynamical systems (see, e.g., [ 7 , 19 , 20 ]), numerical approximation [ 21 , 22 ], and statistical learning [ 17 , 23 ].…”

Section: Introductionmentioning

confidence: 99%

Data-Driven Model Reduction for Stochastic Burgers Equations

2020

Entropy

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We present a class of efficient parametric closure models for 1D stochastic Burgers equations. Casting it as statistical learning of the flow map, we derive the parametric form by representing the unresolved high wavenumber Fourier modes as functionals of the resolved variable’s trajectory. The reduced models are nonlinear autoregression (NAR) time series models, with coefficients estimated from data by least squares. The NAR models can accurately reproduce the energy spectrum, the invariant densities, and the autocorrelations. Taking advantage of the simplicity of the NAR models, we investigate maximal space-time reduction. Reduction in space dimension is unlimited, and NAR models with two Fourier modes can perform well. The NAR model’s stability limits time reduction, with a maximal time step smaller than that of the K-mode Galerkin system. We report a potential criterion for optimal space-time reduction: the NAR models achieve minimal relative error in the energy spectrum at the time step, where the K-mode Galerkin system’s mean Courant–Friedrichs–Lewy (CFL) number agrees with that of the full model.

show abstract

Modeling of missing dynamical systems: deriving parametric models using a nonparametric framework

Cited by 19 publications

References 42 publications

Data-driven Model Reduction for Stochastic Burgers Equations

Data-driven Model Reduction for Stochastic Burgers Equations

Estimating linear response statistics using orthogonal polynomials: An rkhs formulation

Data-Driven Model Reduction for Stochastic Burgers Equations

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