  We show how a complete mathematical description of a complicated physical
phenomenon can be learned from observational data via a hybrid approach
combining three simple and general ingredients: physical assumptions of
smoothness, locality, and symmetry, a weak formulation of differential
equations, and sparse regression. To illustrate this, we extract a system of
governing equations describing flows of incompressible Newtonian fluids -- the
Navier-Stokes equation, the continuity equation, and the boundary conditions --
from numerical data describing a highly turbulent channel flow in three
dimensions. These relations have the familiar form of partial differential
equations, which are easily interpretable and readily provide information about
the relative importance of different physical effects as well as insight into
the quality of the data, serving as a useful diagnostic tool. The approach
described here is remarkably robust, yielding accurate results for very high
noise levels, and should thus be well-suited to experimental data.
