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

Abstract

There is an opportunity for deep learning to revolutionize science and technology by revealing its findings in a human interpretable manner. We develop a novel data-driven approach for creating a human-machine partnership to accelerate scientific discovery. By collecting physical system responses, under carefully selected excitations, we train rational neural networks to learn Green's functions of hidden partial differential equation. These solutions 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 this technique on several examples and capture a range of physics, including advection-diffusion, viscous shocks, and Stokes flow in a lid-driven cavity.Deep learning (DL) holds promise as a scientific tool for discovering elusive patterns within the natural and technological world (1, 2). These patterns hint at undiscovered partial differential equations (PDEs) that describe governing phenomena within biology, fluid dynamics, and physics. From sparse and noisy laboratory observations, one aims to learn mechanistic laws 1

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