  We develop a numerical method to reconstruct systems of ordinary differential
equations (ODEs) from time series data without {\it a priori} knowledge of the
underlying ODEs using sparse basis learning and sparse function reconstruction.
We show that employing sparse representations provides more accurate ODE
reconstruction compared to least-squares reconstruction techniques for a given
amount of time series data. We test and validate the ODE reconstruction method
on known 1D, 2D, and 3D systems of ODEs. The 1D system possesses two stable
fixed points; the 2D system possesses an oscillatory fixed point with closed
orbits; and the 3D system displays chaotic dynamics on a strange attractor. We
determine the amount of data required to achieve an error in the reconstructed
functions to less than $0.1\%$. For the reconstructed 1D and 2D systems, we are
able to match the trajectories from the original ODEs even at long times. For
the 3D system with chaotic dynamics, as expected, the trajectories from the
original and reconstructed systems do not match at long times, but the
reconstructed and original models possess similar Lyapunov exponents. Now that
we have validated this ODE reconstruction method on known models, it can be
employed in future studies to identify new systems of ODEs using time series
data from deterministic systems for which there is no currently known ODE
model.
