Sparsity-promoting algorithms for the discovery of informative Koopman-invariant subspaces

Pan, Shaowu; Arnold-Medabalimi, Nicholas; Duraisamy, Karthik

doi:10.1017/jfm.2021.271

Cited by 37 publications

(10 citation statements)

References 77 publications

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“…4, allowing the user to select which among the models best matches the desired time-series behavior. Pan et al [60] employ a similar manual examination of an FoM followed by refitting, to downselect from a library of eigenvectors in the context of Koopman operators. This method (do one complete STLSQ run and record the models at each step) is similar to how Lasso is done, and is an alternative to sweeping the sparsity parameter λ with multiple complete SINDy runs.…”

Section: ) Figure-of-merit Historiesmentioning

confidence: 99%

A Toolkit for Data-Driven Discovery of Governing Equations in High-Noise Regimes

Delahunt

Kutz

2022

IEEE Access

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We consider the data-driven discovery of governing equations from time-series data in the limit of high noise. The algorithms developed describe an extensive toolkit of methods for circumventing the deleterious effects of noise in the context of the sparse identification of nonlinear dynamics (SINDy) framework. We offer two primary contributions, both focused on noisy data acquired from a system ẋ = f (x). First, we propose, for use in high-noise settings, an extensive toolkit of critically enabling extensions for the SINDy regression method, to progressively cull functionals from an over-complete library and yield a set of sparse equations that regress to the derivate ẋ. This toolkit includes: (regression step) weight timepoints based on estimated noise, use ensembles to estimate coefficients, and regress using FFTs; (culling step) leverage linear dependence of functionals, and restore and protect culled functionals based on Figures of Merit (FoMs). In a novel Assessment step, we define FoMs that compare model predictions to the original time-series (i.e., x(t) rather than ẋ(t)). These innovations can extract sparse governing equations and coefficients from high-noise time-series data (e.g., 300% added noise). For example, it discovers the correct sparse libraries in the Lorenz system, with median coefficient estimate errors equal to 1%−3% (for 50% noise), 6%−8% (for 100% noise); and 23%−25% (for 300% noise). The enabling modules in the toolkit are combined into a single method, but the individual modules can be tactically applied in other equation discovery methods (SINDy or not) to improve results on high-noise data. Second, we propose a technique, applicable to any model discovery method based on ẋ = f (x), to assess the accuracy of a discovered model in the context of non-unique solutions due to noisy data. Currently, this non-uniqueness can obscure a discovered model's accuracy and thus a discovery method's effectiveness. We describe a technique that uses linear dependencies among functionals to transform a discovered model into an equivalent form that is closest to the true model, enabling more accurate assessment of a discovered model's accuracy.

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Section: ) Figure-of-merit Historiesmentioning

confidence: 99%

A Toolkit for Data-Driven Discovery of Governing Equations in High-Noise Regimes

Delahunt

Kutz

2022

IEEE Access

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“…Hence, computing the spectrum of matrix K by direct SVD is computationally intensive, even though most of the eigenvalues will be zero. Method-of-snapshot, parallel version of SVD, or randomized SVD can be used to attack this difficulty [10,11]. In this project, we use a more simple randomized method by generalizing the DMD algorithm.…”

Section: Dynamic Mode Decompositionmentioning

confidence: 99%

“…A commonly used projection matrix is based on SVD of the input matrix X = UΣV * and the projection matrix is chosen to be P = U * , here, * represents the conjugate transpose of a matrix. Using Equation (11) and SVD of X, the operator on the projected space can be formulated as K = U * YVΣ −1 .…”

Section: Remark 2 (Projection By Svd)mentioning

confidence: 99%

Randomized Projection Learning Method for Dynamic Mode Decomposition

Surasinghe

Bollt

2021

Mathematics

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A data-driven analysis method known as dynamic mode decomposition (DMD) approximates the linear Koopman operator on a projected space. In the spirit of Johnson–Lindenstrauss lemma, we will use a random projection to estimate the DMD modes in a reduced dimensional space. In practical applications, snapshots are in a high-dimensional observable space and the DMD operator matrix is massive. Hence, computing DMD with the full spectrum is expensive, so our main computational goal is to estimate the eigenvalue and eigenvectors of the DMD operator in a projected domain. We generalize the current algorithm to estimate a projected DMD operator. We focus on a powerful and simple random projection algorithm that will reduce the computational and storage costs. While, clearly, a random projection simplifies the algorithmic complexity of a detailed optimal projection, as we will show, the results can generally be excellent, nonetheless, and the quality could be understood through a well-developed theory of random projections. We will demonstrate that modes could be calculated for a low cost by the projected data with sufficient dimension.

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“…Linear subspaces, however, are highly restrictive and ill-suited to handle parametric dependencies. Attempts to circumvent these shortcomings include using multiple linear subspaces covering different temporal or spatial domains [9,10,11,12], multi-resolution DMD [13], diffusion map embeddings [14,15,16,17], or more recently, using deep learning to compute underlying nonlinear subspaces which are advantageous for dynamics, both linear and nonlinear [18,19,20,21,22]. These techniques represent data-driven architectures for extracting order-parameter descriptions of the underlying spatio-temporal dynamics observed [23].…”

Section: Introductionmentioning

confidence: 99%

Data-driven discovery of governing equations for coarse-grained heterogeneous network dynamics

Owens¹,

Kutz²

2022

Preprint

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We leverage data-driven model discovery methods to determine the governing equations for the emergent behavior of heterogeneous networked dynamical systems. Specifically, we consider networks of coupled nonlinear oscillators whose collective behaviour approaches a limit cycle. Stable limit-cycles are of interest in many biological applications as they model self-sustained oscillations (e.g. heart beats, chemical oscillations, neurons firing, circadian rhythm). For systems that display relaxation oscillations, our method automatically detects boundary (time) layer structures in the dynamics, fitting inner and outer solutions and matching them in a data-driven manner. We demonstrate the method on well-studied systems: the Rayleigh Oscillator and the Van der Pol Oscillator. We then apply the mathematical framework to discovering lowdimensional dynamics in networks of semi-synchronized Kuramoto, Rayleigh, Rossler, and Fitzhugh-Nagumo oscillators, as well as heterogeneous combinations thereof. We also provide a numerical exploration of the dimension of collective network dynamics as a function of several network parameters, showing that the discovery of coarse-grained variables and dynamics can be accomplished with the proposed architecture.

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Sparsity-promoting algorithms for the discovery of informative Koopman-invariant subspaces

Abstract: Abstract

Cited by 37 publications

References 77 publications

A Toolkit for Data-Driven Discovery of Governing Equations in High-Noise Regimes

A Toolkit for Data-Driven Discovery of Governing Equations in High-Noise Regimes

Randomized Projection Learning Method for Dynamic Mode Decomposition

Data-driven discovery of governing equations for coarse-grained heterogeneous network dynamics

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