Kernel Methods for the Approximation of Nonlinear Systems

Bouvrie, Jake; Hamzi, Boumediene

doi:10.1137/14096815x

Cited by 51 publications

(55 citation statements)

References 41 publications

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“…In this section, we give a brief review of reproducing kernel Hilbert spaces (RKHSs) as used in statistical learning theory. The discussion here mostly follows previous studies; for a historical perspective, see also Aronszajn and Schönberg, and for a recent application related to dynamics, see Bouvrie and Hamzi . The key problem we want to study with this theory is to distinguish two different probability distributions.…”

Section: Kernels and MMDmentioning

confidence: 98%

A note on kernel methods for multiscale systems with critical transitions

Hamzi

Kuehn

Mohamed

2018

Math Methods in App Sciences

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We study the maximum mean discrepancy (MMD) in the context of critical transitions modelled by fast-slow stochastic dynamical systems. We establish a new link between the dynamical theory of critical transitions with the statistical aspects of the MMD. In particular, we show that a formal approximation of the MMD near fast subsystem bifurcation points can be computed to leading order. This leading order approximation shows that the MMD depends intricately on the fast-slow systems parameters, which can influence the detection of potential early-warning signs before critical transitions. However, the MMD turns out to be an excellent binary classifier to detect the change-point location induced by the critical transition. We cross-validate our results by numerical simulations for a van der Pol-type model. KEYWORDSbifurcation, critical transition, kernel methods, maximum mean discrepancy, multiscale system, time series, tipping point• Change-point detection: In a time series generated by a fast-slow SODE, there could be many different types and sizes of drastic jumps. Hence, it would not only be useful to develop an automatic and generic classifiers, 10-12 when we actually observe a critical transition, but also to cross validate a classifier against explicit low-dimensional models.Math Meth Appl Sci. 2019;42:907-917. wileyonlinelibrary.com/journal/mma

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Section: Kernels and MMDmentioning

confidence: 98%

A note on kernel methods for multiscale systems with critical transitions

Hamzi

Kuehn

Mohamed

2018

Math Methods in App Sciences

Self Cite

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“…Other choices are for example: covariance-weighted products for Gaussian-noise-driven systems yielding system covariances [76], reproducing kernel Hilbert spaces (RKHS) [77], such as the polynomial, Gaussian or Sigmoid kernels [78], or energy-stable inner products [79]. Also, weighted Gramians [36] and time-weighted system Gramians [80] can be computed using this interface, i.e., dp = @(x,y) mtimes([0:h:T].ˆk.…”

Section: Inner Product Interfacementioning

confidence: 99%

emgr—The Empirical Gramian Framework

Himpe

2018

Algorithms

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System Gramian matrices are a well-known encoding for properties of input-output systems such as controllability, observability or minimality. These so-called system Gramians were developed in linear system theory for applications such as model order reduction of control systems. Empirical Gramians are an extension to the system Gramians for parametric and nonlinear systems as well as a data-driven method of computation. The empirical Gramian framework -emgrimplements the empirical Gramians in a uniform and configurable manner, with applications such as Gramian-based (nonlinear) model reduction, decentralized control, sensitivity analysis, parameter identification and combined state and parameter reduction. Keywords

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“…Therefore, cheaper numerical approximations using "adequate-fidelity" models are usually acceptable [2]. In this regard, reduced order modeling offers a viable technique to address systems characterized by underlying patterns [3][4][5][6][7][8][9][10][11][12]. This is especially true for fluid flows dominated by coherent structures (e.g., atmospheric and oceanic flows) [13][14][15][16][17][18][19][20][21].…”

Section: Introductionmentioning

confidence: 99%

A long short-term memory embedding for hybrid uplifted reduced order models

Ahmed

San

Rasheed

et al. 2020

Physica D: Nonlinear Phenomena

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In this paper, we introduce an uplifted reduced order modeling (UROM) approach through the integration of standard projection based methods with long short-term memory (LSTM) embedding. Our approach has three modeling layers or components. In the first layer, we utilize an intrusive projection approach to model dynamics represented by the largest modes. The second layer consists of an LSTM model to account for residuals beyond this truncation. This closure layer refers to the process of including the residual effect of the discarded modes into the dynamics of the largest scales. However, the feasibility of generating a low rank approximation tails off for higher Kolmogorov n-width systems due to the underlying nonlinear processes. The third uplifting layer, called superresolution, addresses this limited representation issue by expanding the span into a larger number of modes utilizing the versatility of LSTM. Therefore, our model integrates a physics-based projection model with a memory embedded LSTM closure and an LSTM based super-resolution model. In several applications, we exploit the use of Grassmann manifold to construct UROM for unseen conditions. We performed numerical experiments by using the Burgers and Navier-Stokes equations with quadratic nonlinearity. Our results show robustness of the proposed approach in building reduced order models for parameterized systems and confirm the improved trade-off between accuracy and efficiency.Keywords Hybrid analysis and modeling · Supervised machine learning · Long short-term memory · Model reduction · Galerkin projection · Grassmann manifold.Reduced order models (ROMs) have shown great success for prototypical problems in different fields. In particular, Galerkin projection (GP) coupled with proper orthogonal decomposition (POD) capability to extract the most energetic modes has been used to build ROMs for linear and nonlinear systems [22][23][24][25][26][27][28][29]. These ROMs preserve sufficient arXiv:1912.06756v1 [physics.flu-dyn]

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Kernel Methods for the Approximation of Nonlinear Systems

Cited by 51 publications

References 41 publications

A note on kernel methods for multiscale systems with critical transitions

A note on kernel methods for multiscale systems with critical transitions

emgr—The Empirical Gramian Framework

A long short-term memory embedding for hybrid uplifted reduced order models

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