  In spatially extended systems, it is common to find latent variables that are
hard, or even impossible, to measure with acceptable precision, but are
crucially important for the proper description of the dynamics. This
substantially complicates construction of an accurate model for such systems
using data-driven approaches. The present paper illustrates how physical
constraints can be employed to overcome this limitation using the example of a
weakly turbulent quasi-two-dimensional Kolmogorov flow driven by a steady
Lorenz force with an unknown spatial profile. Specifically, the terms involving
latent variables in the partial differential equations governing the dynamics
can be eliminated at the expense of raising the order of that equation. We show
that local polynomial interpolation combined with symbolic regression can
handle sparse data on grids that are representative of typical experimental
measurement techniques such as particle image velocimetry. However, we also
find that the reconstructed model is sensitive to measurement noise and trace
this sensitivity to the presence of high order spatial and/or temporal
derivatives.
