Abstract: A major challenge in the study of dynamical systems is that of model discovery: turning data into models that are not just predictive, but provide insight into the nature of the underlying dynamical system that generated the data. This problem is made more difficult by the fact that many systems of interest exhibit parametric dependencies and diverse behaviors across multiple time scales. We introduce a number of data-driven strategies for discovering nonlinear dynamical systems, their coordinates and their control laws from data. We consider two canonical cases: (i) systems for which we have full measurements of the governing variables, and (ii) systems for which we have incomplete measurements. For systems with full state measurements, we show that the recent sparse identification of nonlinear dynamical systems (SINDy) method can discover governing equations with relatively little data and introduce a sampling method that allows SINDy to scale efficiently to problems with multiple time scales and parametric dependencies. We can also regress to data-driven control laws that are capable of learning how to control a given system. Together, our approaches provide a suite of mathematical strategies for reducing the data required to discover, model and control nonlinear systems. The methods are demonstrated on optical fiber lasers and meta-material antennas.
