Kernel learning for robust dynamic mode decomposition: linear and nonlinear disambiguation optimization

Baddoo, Peter J.; Herrmann, Benjamín; McKeon, Beverley; Brunton, Steven L.

doi:10.1098/rspa.2021.0830

Cited by 36 publications

(25 citation statements)

References 69 publications

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“…where ε(t) denotes the error associated with the projection of s(t) − s ref onto the linear subspace V. While many different manifold constructs are conceivable, we focus on a polynomial mapping between the highdimensional data samples and their lower-dimensional representations. Explicit nonlinear mappings with a polynomial structure were originally proposed in manifold learning [42], but similar formulations have emerged recently in model reduction [2,3,5,43,44]. Our attention will be restricted to the case of quadratic Kronecker products:…”

Section: Data-driven Quadratic Solution-manifoldsmentioning

confidence: 99%

Operator inference for non-intrusive model reduction with quadratic manifolds

Geelen¹,

Wright²,

Willcox³

2022

Preprint

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This paper proposes a novel approach for learning a data-driven quadratic manifold from highdimensional data, then employing the quadratic manifold to derive efficient physics-based reduced-order models. The key ingredient of the approach is a polynomial mapping between high-dimensional states and a low-dimensional embedding. This mapping comprises two parts: a representation in a linear subspace (computed in this work using the proper orthogonal decomposition) and a quadratic component. The approach can be viewed as a form of data-driven closure modeling, since the quadratic component introduces directions into the approximation that lie in the orthogonal complement of the linear subspace, but without introducing any additional degrees of freedom to the low-dimensional representation. Combining the quadratic manifold approximation with the operator inference method for projectionbased model reduction leads to a scalable non-intrusive approach for learning reduced-order models of dynamical systems. Applying the new approach to transport-dominated systems of partial differential equations illustrates the gains in efficiency that can be achieved over approximation in a linear subspace.

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Section: Data-driven Quadratic Solution-manifoldsmentioning

confidence: 99%

Operator inference for non-intrusive model reduction with quadratic manifolds

Geelen¹,

Wright²,

Willcox³

2022

Preprint

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“…The leading numerical algorithm used to approximate K from trajectory data is dynamic mode decomposition (DMD) (Tu et al 2014;Kutz et al 2016a). First introduced by Schmid (2009Schmid ( , 2010 in the fluids community and connected to the Koopman operator in Rowley et al (2009), there are now numerous extensions and variants of DMD (Chen, Tu & Rowley 2012;Wynn et al 2013;Jovanović, Schmid & Nichols 2014;Kutz, Fu & Brunton 2016b;Noack et al 2016;Korda & Mezić 2018;Loiseau & Brunton 2018;Deem et al 2020;Klus et al 2020;Baddoo et al 2021Baddoo et al , 2022Herrmann et al 2021;Wang & Shoele 2021;. Of particular interest is extended DMD (EDMD) (Williams, Kevrekidis & Rowley 2015a), which generalised DMD to a broader class of nonlinear observables.…”

Section: Introductionmentioning

confidence: 99%

Residual dynamic mode decomposition: robust and verified Koopmanism

2023

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Dynamic mode decomposition (DMD) describes complex dynamic processes through a hierarchy of simpler coherent features. DMD is regularly used to understand the fundamental characteristics of turbulence and is closely related to Koopman operators. However, verifying the decomposition, equivalently the computed spectral features of Koopman operators, remains a significant challenge due to the infinite-dimensional nature of Koopman operators. Challenges include spurious (unphysical) modes and dealing with continuous spectra, which both occur regularly in turbulent flows. Residual dynamic mode decomposition (ResDMD), introduced by Colbrook & Townsend (Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems. 2021. arXiv:2111.14889), overcomes such challenges through the data-driven computation of residuals associated with the full infinite-dimensional Koopman operator. ResDMD computes spectra and pseudospectra of general Koopman operators with error control and computes smoothed approximations of spectral measures (including continuous spectra) with explicit high-order convergence theorems. ResDMD thus provides robust and verified Koopmanism. We implement ResDMD and demonstrate its application in various fluid dynamic situations at varying Reynolds numbers from both numerical and experimental data. Examples include vortex shedding behind a cylinder, hot-wire data acquired in a turbulent boundary layer, particle image velocimetry data focusing on a wall-jet flow and laser-induced plasma acoustic pressure signals. We present some advantages of ResDMD: the ability to resolve nonlinear and transient modes verifiably; the verification of learnt dictionaries; the verification of Koopman mode decompositions; and spectral calculations with reduced broadening effects. We also discuss how a new ordering of modes via residuals enables greater accuracy than the traditional modulus ordering (e.g. when forecasting) with a smaller dictionary. This result paves the way for more significant dynamic compression of large datasets without sacrificing accuracy.

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“…In this regard, the so-called kernel methods are also considered, when a certain set, called the dictionary, of transformations is selected to reduce the dimension of the basis arising in the spectrum approximation of the infinite-dimensional Koopman operator. For instance, in 29 a robust kernel method is presented, which allowed to separate with good accuracy the linear and nonlinear dynamics for dynamical systems of several different types.…”

Section: Introductionmentioning

confidence: 99%

Ffowcs Williams – Hawkings analogy for near-field acoustic sources analysis

Karakulev

Kozubskaya

Plaksin

et al. 2022

International Journal of Aeroacoustics

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The paper expands the scope of applying the Ffowcs Williams – Hawkings integration method. We propose using the acoustic field generated from time-dependent data stored on the FW-H control surface as the same common field for computational acoustic beamforming and dynamic mode decomposition methods to analyze the aerodynamic noise sources. We exemplify that it leads to obtaining mutually consistent and complementary information for reliable prediction of acoustic sources characteristics in the process of inverting data produced by a CFD simulation. Moreover, as the results of applying computational acoustic beamforming and dynamic mode decomposition methods depend on many geometric and algorithmic inputs, the proposed approach makes it possible to use various sets of the latter for a comprehensive analysis of obtained inversions and to form the final answer by an averaging procedure. We illustrate this by taking advantage of fast generating the examined acoustic field snapshots in any required region by the FW-H integration method for the recently developed new inverse computational acoustic beamforming algorithm and the standard dynamic mode decomposition method when carrying out a sensitivity study of the predicted acoustic source. The capabilities of the developed approach are demonstrated on the data of CFD scale-resolving simulation of turbulent flow over the 30P30N high-lift configuration.

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Kernel learning for robust dynamic mode decomposition: linear and nonlinear disambiguation optimization

Cited by 36 publications

References 69 publications

Operator inference for non-intrusive model reduction with quadratic manifolds

Operator inference for non-intrusive model reduction with quadratic manifolds

Residual dynamic mode decomposition: robust and verified Koopmanism

Ffowcs Williams – Hawkings analogy for near-field acoustic sources analysis

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