From data to dynamics: discovering governing equations from data
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Governing laws and equations, such as Newton's second law for classical mechanics and the Navier-Stokes equations thence derived, have been responsible throughout history for numerous scientific breakthroughs in the physical and engineering sciences. There are many systems of interest for which large quantities of measurement data have been collected, but the underlying governing equations remain unknown. While machine learning approaches such as sparse regression and deep neural networks have been successful at discovering governing laws and reduced models from data, many challenges still remain. In this work, we focus on the discovery of nonlinear dynamical systems models from data. We present several methods based on the sparse identification of nonlinear dynamics (SINDy) algorithm. These approaches address a number of challenges that occur when dealing with scientific data sets, including unknown coordinates, multiscale dynamics, parametric dependencies, and outliers. Our methods focus on discovering parsimonious models, as parsimony is key for obtaining models that have physical interpretations and can generalize to predict previously unobserved behaviors.
- Applied mathematics