Compressive Sensing and Low-Rank Libraries for Classification of Bifurcation Regimes in Nonlinear Dynamical Systems

By Steven L. Brunton, Jonathan H. Tu, Ido Bright, J. Nathan Kutz. 2013

125 citations recorded by the source corpus.

We show that for complex nonlinear systems, model reduction and compressive sensing strategies can be combined to great advantage for classifying, projecting, and reconstructing the relevant low-dimensional dynamics. ${L}_2$-based dimensionality reduction methods such as the proper orthogonal decomposition are used to construct separate modal libraries and Galerkin models based on data from a number of bifurcation regimes. These libraries are then concatenated into an overcomplete library, and ${L}_1$-sparse representation in this library from a few noisy measurements results in correct identification of the bifurcation regime. This technique provides an objective and general framework for classifying the bifurcation parameters and, therefore, the underlying dynamics and stability. After classifying the bifurcation regime, it is possible to employ a low-dimensional Galerkin model, only on modes relevant to that bifurcation value. These methods are demonstrated on the complex Ginzburg--Landau equation usin...

Subfields: Atomic and Molecular Physics, and Optics, Computer Networks and Communications, Statistical and Nonlinear Physics

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Citations in the corpus

Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations (2018, methodology): “More recently, sparsity [23] has been used to determine the governing dynamical system [24, 25, 26, 27, 28, 29, 30, 31, 32, 33].”
Koopman Invariant Subspaces and Finite Linear Representations of Nonlinear Dynamical Systems for Control (2015, background): “This follows a growing trend to exploit sparsity in dynamics [35, 36, 37] and dynamical systems [38, 39, 40].”
Multiresolution Dynamic Mode Decomposition (2015, background): “In the dynamical applications considered here, one might exploit these patterns, or DMD modes, by building libraries of low-rank dynamical modes, much like is done with POD modes [25, 26, 27].”
Chaos as an intermittently forced linear system (2016, background): “• Chaos is important: Dynamics [10], Lorenz [14] • Takens embedding is profound and important [23] • Connection to Koopman analysis [12] • Connection to identifying dynamics and DMD [4, 13] • Koopman invariant subspace [?] • Time series classics: review [1], book [25] • ERA [11], SSA [3], NLSA [9]”
Sparse identification of nonlinear dynamics for model predictive control in the low-data limit (2017, background): “A heterogeneous model structure can potentially be incorporated into SINDy by additionally learning a library of models [74,75].”
On the Compressive Spectral Method (2014, background): “In [25], low-rank libraries are used to sparsely approximate solutions to dynamical systems and thereby identify bifurcation regimes.”