Publication 18-CNA-012
Extracting structured dynamical systems using sparse optimization with very few samples
Hayden Schaeffer
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
schaeffer@cmu.edu
Giang Tran
Department of Applied Mathematics
University of Waterloo
Waterloo, Ontario, Canada
giang.tran@uwaterloo.ca
Rachel Ward
Department of Mathematics
The University of Texas at Austin
Austin, TX
rward@math.utexas.edu
Linan Zhang
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
linanz@andrew.cmu.edu
Abstract: Learning governing equations allows for deeper understanding of the structure and dynamics of data. We present a random sampling method for learning structured dynamical systems from under-sampled and possibly noisy state-space measurements. The learning problem takes the form of a sparse least-squares fitting over a large set of candidate functions. Based on a Bernstein-like inequality for partly dependent random variables, we provide theoretical guarantees on the recovery rate of the sparse coefficients and the identification of the candidate functions for the corresponding problem. Computational results are demonstrated on datasets generated by the Lorenz 96 equation, the viscous Burgers' equation, and the two-component reaction-diffusion equations (which is challenging due to parameter sensitives in the model). This formulation has several advantages including ease of use, theoretical guarantees of success, and computational efficiency with respect to ambient dimension and number of candidate functions.
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