  Machine learning offers an intriguing alternative to first-principles
analysis for discovering new physics from experimental data. However, to date,
purely data-driven methods have only proven successful in uncovering physical
laws describing simple, low-dimensional systems with low levels of noise. Here
we demonstrate that combining a data-driven methodology with some general
physical principles enables discovery of a quantitatively accurate model of a
non-equilibrium spatially-extended system from high-dimensional data that is
both noisy and incomplete. We illustrate this using an experimental weakly
turbulent fluid flow where only the velocity field is accessible. We also show
that this hybrid approach allows reconstruction of the inaccessible variables
-- the pressure and forcing field driving the flow.
