Discovery of Nonlinear Multiscale Systems: Sampling Strategies and Embeddings Article Swipe
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· 2019
· Open Access
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· DOI: https://doi.org/10.1137/18m1188227
· OA: W2803477776
A major challenge in the study of dynamical systems is that of model\ndiscovery: turning data into models that are not just predictive, but provide\ninsight into the nature of the underlying dynamical system that generated the\ndata. This problem is made more difficult by the fact that many systems of\ninterest exhibit diverse behaviors across multiple time scales. We introduce a\nnumber of data-driven strategies for discovering nonlinear multiscale dynamical\nsystems and their embeddings from data. We consider two canonical cases: (i)\nsystems for which we have full measurements of the governing variables, and\n(ii) systems for which we have incomplete measurements. For systems with full\nstate measurements, we show that the recent sparse identification of nonlinear\ndynamical systems (SINDy) method can discover governing equations with\nrelatively little data and introduce a sampling method that allows SINDy to\nscale efficiently to problems with multiple time scales. Specifically, we can\ndiscover distinct governing equations at slow and fast scales. For systems with\nincomplete observations, we show that the Hankel alternative view of Koopman\n(HAVOK) method, based on time-delay embedding coordinates, can be used to\nobtain a linear model and Koopman invariant measurement system that nearly\nperfectly captures the dynamics of nonlinear quasiperiodic systems. We\nintroduce two strategies for using HAVOK on systems with multiple time scales.\nTogether, our approaches provide a suite of mathematical strategies for\nreducing the data required to discover and model nonlinear multiscale systems.\n
