Model discovery in the sparse sampling regime

TL;DR

This paper demonstrates that deep learning techniques, especially physics-informed neural networks, can effectively discover underlying partial differential equations from sparsely and irregularly sampled data, outperforming traditional methods.

Contribution

It introduces a deep learning approach using physics-informed neural networks for model discovery from off-grid, sparse, and noisy data, improving upon classical interpolation and differentiation methods.

Findings

Deep learning improves PDE model discovery from sparse data.

Physics-informed neural networks outperform spline interpolation.

Method successfully identifies physical processes in synthetic and real data.

Abstract

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.