Data-driven discovery of Green's functions with human-understandable deep learning

TL;DR

This paper introduces a data-driven deep learning method that learns Green's functions of linear PDEs from physical responses, providing human-understandable insights into complex physical systems.

Contribution

It develops a novel approach using rational neural networks trained on Gaussian process responses to uncover interpretable Green's functions of hidden PDEs.

Findings

Successfully captures conservation laws and symmetries.

Identifies shock locations and boundary effects.

Applies to diverse physics problems like advection-diffusion and Stokes flow.

Abstract

There is an opportunity for deep learning to revolutionize science and technology by revealing its findings in a human interpretable manner. To do this, we develop a novel data-driven approach for creating a human-machine partnership to accelerate scientific discovery. By collecting physical system responses under excitations drawn from a Gaussian process, we train rational neural networks to learn Green's functions of hidden linear partial differential equations. These functions reveal human-understandable properties and features, such as linear conservation laws and symmetries, along with shock and singularity locations, boundary effects, and dominant modes. We illustrate the technique on several examples and capture a range of physics, including advection-diffusion, viscous shocks, and Stokes flow in a lid-driven cavity.