Data-driven discovery of partial differential equation models with latent variables

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 sparse regression can handle data on spatiotemporal 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.