Model discovery in the sparse sampling regime

To improve the physical understanding and the predictions of complex dynamic systems, such as ocean dynamics and weather predictions, it is of paramount interest to identify interpretable models from coarsely and off-grid sampled observations. In this work we investigate how deep learning can improve model discovery of partial differential equations when the spacing between sensors is large and the samples are not placed on a grid. We show how leveraging physics informed neural network interpolation and automatic differentiation, allow to better fit the data and its spatiotemporal derivatives, compared to more classic spline interpolation and numerical differentiation techniques. As a result, deep learning based model discovery allows to recover the underlying equations, even when sensors are placed further apart than the data's characteristic length scale and in the presence of high noise levels. We illustrate our claims on both synthetic and experimental data sets where combinations of physical processes such as (non)-linear advection, reaction and diffusion are correctly identified. Mathematical models are central in modelling complex dynamical processes such as climate change, the spread of an epidemic or to design aircrafts.