Robust identifiability for symbolic recovery of differential equations

TL;DR

This paper develops a mathematical framework and algorithms to analyze and ensure the identifiability of physical laws governed by PDEs in noisy data environments, addressing a key challenge in data-driven discovery of differential equations.

Contribution

It introduces a novel framework and algorithms for assessing the uniqueness of PDE-based models under noise, extending prior work to more realistic noisy data scenarios.

Findings

Algorithms effectively detect PDE uniqueness with noise

Thresholds established for noise levels impacting identifiability

Numerical experiments validate the approach

Abstract

Recent advancements in machine learning have transformed the discovery of physical laws, moving from manual derivation to data-driven methods that simultaneously learn both the structure and parameters of governing equations. This shift introduces new challenges regarding the validity of the discovered equations, particularly concerning their uniqueness and, hence, identifiability. While the issue of non-uniqueness has been well-studied in the context of parameter estimation, it remains underexplored for algorithms that recover both structure and parameters simultaneously. Early studies have primarily focused on idealized scenarios with perfect, noise-free data. In contrast, this paper investigates how noise influences the uniqueness and identifiability of physical laws governed by partial differential equations (PDEs). We develop a comprehensive mathematical framework to analyze the uniqueness of PDEs in the presence of noise and introduce new algorithms that account for noise, providing thresholds to assess uniqueness and identifying situations where excessive noise hinders reliable conclusions. Numerical experiments demonstrate the effectiveness of these algorithms in detecting uniqueness despite the presence of noise.