Discovery of partial differential equations from highly noisy and sparse data with physics-informed information criterion

TL;DR

This paper introduces a physics-informed information criterion (PIC) that robustly identifies PDEs from noisy and sparse data, enabling discovery of governing equations in complex physical systems.

Contribution

The paper presents a novel PIC method that improves PDE discovery accuracy and robustness in noisy, sparse data scenarios, and applies it to real physical data for the first time.

Findings

PIC achieves state-of-the-art robustness to noise and sparsity.

PIC successfully discovers unrevealed macroscale PDEs from microscopic data.

Discovered PDEs are accurate, parsimonious, and respect physical symmetries.

Abstract

Data-driven discovery of PDEs has made tremendous progress recently, and many canonical PDEs have been discovered successfully for proof-of-concept. However, determining the most proper PDE without prior references remains challenging in terms of practical applications. In this work, a physics-informed information criterion (PIC) is proposed to measure the parsimony and precision of the discovered PDE synthetically. The proposed PIC achieves state-of-the-art robustness to highly noisy and sparse data on seven canonical PDEs from different physical scenes, which confirms its ability to handle difficult situations. The PIC is also employed to discover unrevealed macroscale governing equations from microscopic simulation data in an actual physical scene. The results show that the discovered macroscale PDE is precise and parsimonious, and satisfies underlying symmetries, which facilitates understanding and simulation of the physical process. The proposition of PIC enables practical applications of PDE discovery in discovering unrevealed governing equations in broader physical scenes.