Discovery of Partial Differential Equations from Highly Noisy and Sparse Data with Physics-Informed Information...

Area of Science:

• Computational Physics
• Applied Mathematics
• Data Science

Background:

• Data-driven discovery of partial differential equations (PDEs) shows promise but struggles with selecting the most appropriate equation in practical scenarios.
• Existing methods lack robust metrics for evaluating the parsimony and precision of discovered PDEs, especially with imperfect data.

Purpose of the Study:

• To introduce a novel Physics-Informed Information Criterion (PIC) for quantitatively assessing the quality of discovered PDEs.
• To demonstrate the robustness and applicability of PIC in challenging data conditions and real-world scientific discovery.

Main Methods:

• Development of the Physics-Informed Information Criterion (PIC) to measure PDE parsimony and precision.
• Validation of PIC using seven canonical PDEs across diverse physical domains with noisy and sparse data.
• Application of PIC to uncover macroscale governing equations from microscopic simulation data in a specific physical system.

Main Results:

• PIC demonstrates significant robustness against highly noisy and sparse data for canonical PDEs.
• The criterion effectively evaluates the parsimony and precision of discovered equations.
• PIC facilitated the discovery of a precise, parsimonious macroscale PDE from microscopic data, respecting underlying symmetries.

Conclusions:

• The proposed PIC is a reliable tool for selecting accurate and concise PDEs in data-driven discovery.
• PIC enhances the practical applicability of PDE discovery, enabling the identification of unknown governing equations in complex physical systems.
• This work advances the field by providing a crucial metric for evaluating and discovering fundamental scientific laws from data.