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

## TL;DR

This paper presents a data-driven method to identify PDE models with latent variables in spatial systems, using physical constraints and symbolic regression to handle sparse data, despite noise sensitivity.

## Contribution

It introduces a novel approach combining physical constraints, polynomial interpolation, and symbolic regression to model systems with unmeasured latent variables from sparse data.

## Key findings

- Physical constraints help eliminate latent variables in PDEs.
- Local polynomial interpolation can handle sparse experimental data.
- Reconstructed models are sensitive to measurement noise.

## Abstract

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.