  Discovering nonlinear differential equations that describe system dynamics
from empirical data is a fundamental challenge in contemporary science. Here,
we propose a methodology to identify dynamical laws by integrating denoising
techniques to smooth the signal, sparse regression to identify the relevant
parameters, and bootstrap confidence intervals to quantify the uncertainty of
the estimates. We evaluate our method on well-known ordinary differential
equations with an ensemble of random initial conditions, time series of
increasing length, and varying signal-to-noise ratios. Our algorithm
consistently identifies three-dimensional systems, given moderately-sized time
series and high levels of signal quality relative to background noise. By
accurately discovering dynamical systems automatically, our methodology has the
potential to impact the understanding of complex systems, especially in fields
where data are abundant, but developing mathematical models demands
considerable effort.
