Model discovery in the sparse sampling regime
This addresses the challenge of model discovery in complex dynamic systems like ocean dynamics and weather predictions, where data is sparse and irregularly sampled, representing an incremental improvement over classic interpolation and differentiation techniques.
The paper tackles the problem of identifying interpretable partial differential equation models from coarsely and off-grid sampled observations, showing that deep learning-based model discovery recovers underlying equations even when sensor spacing exceeds the data's characteristic length scale and under high noise levels.
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.