Predicting catastrophes in nonlinear dynamical systems by compressive sensing.

By Wenxu Wang, Rui Yang, Ying-Cheng Lai, V. Kovanis, Celso Grebogi. 2011

328 citations recorded by the source corpus.

An extremely challenging problem of significant interest is to predict catastrophes in advance of their occurrences. We present a general approach to predicting catastrophes in nonlinear dynamical systems under the assumption that the system equations are completely unknown and only time series reflecting the evolution of the dynamical variables of the system are available. Our idea is to expand the vector field or map of the underlying system into a suitable function series and then to use the compressive-sensing technique to accurately estimate the various terms in the expansion. Examples using paradigmatic chaotic systems are provided to demonstrate our idea and potential challenges are discussed.

Subfields: Computational Mechanics, Molecular Biology, 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 (Tibshirani, 1996) has been used to determine the governing dynamical system (Brunton et al., 2016; Mangan et al., 2016; Wang et al., 2011; Schaeffer et al., 2013; Ozoliņš et al., 2013; Mackey et al., 2014; Brunton et al., 2014; Proctor et al., 2014; Bai et al., 2014; Tran an”
Discovering governing equations from data by sparse identification of nonlinear dynamical systems (2015, background): “Proc Natl Acad Sci USA 104(23):9558–9563.”
Network resilience (2020, background): “In addition, the accumulations of massive data and rapid development of computational methods make it possible to identify and predict the exact forms of dynamic models directly from empirical data [49, 50, 51, 52].”
Model selection for dynamical systems via sparse regression and information criteria (2017, background): “Compressed sensing was ﬁrst used to determine the active terms in the dynamics [26], although it does not work well with overdetermined systems that arise when measurements are abundant.”
WEAK SINDY FOR PARTIAL DIFFERENTIAL EQUATIONS (2020, background): “In [22] the authors extended this framework to the context of catastrophe prediction and used compressed sensing techniques to enforce sparsity.”
Data-driven discovery of coordinates and governing equations (2019, background): “Indeed, the data-driven discovery of dynamical systems is the focus of intense research efforts across the physical and engineering sciences [1–15].”
Data Based Identification and Prediction of Nonlinear and Complex Dynamical Systems (2016, methodology): “[23], a number of examples were given in which the bifurcation diagrams computed”
Inferring Biological Networks by Sparse Identification of Nonlinear Dynamics (2016, methodology): “SINDy uses sparse regression [57] for improved numerical robustness in noisy overdetermined problems, as opposed to earlier methods [58] that use compressed sensing [59]–[61].”
Koopman Invariant Subspaces and Finite Linear Representations of Nonlinear Dynamical Systems for Control (2015, methodology): “, genetic programming [48]) to identify dynamics [49, 50]; a similar method has been used to predict catastrophes in dynamical systems [51].”
Constrained sparse Galerkin regression (2016, methodology): “…as the LASSO (Tibshirani 1996) or the sequential thresholded least-squares algorithm used in SINDy, improves the numerical robustness of this identification for noisy overdetermined problems, in contrast to earlier methods (Wang et al. 2011) that used compressed sensing (Candès 2006; Donoho 2006).”
Chaos as an intermittently forced linear system (2016, methodology): “In many cases, we may need to normalize the columns of Θ(X) first to ensure that the restricted isometry property holds [99]; this is especially important when the entries in X are small, since powers of X will be minuscule.”
Sparse Identification of Nonlinear Dynamics with Control (SINDYc) (2016, methodology): “An earlier related algorithm (Wang et al., 2011) uses compressed sensing (Donoho, 2006; Candès, 2006; Baraniuk, 2007), while our algorithm uses sparse regression (Tibshirani, 1996) to handle measurement noise and overdetermined cases when we have more time snapshots than state measurements.”