Robust identifiability for symbolic recovery of differential equations
This addresses the challenge of ensuring reliable symbolic recovery of differential equations for researchers in scientific machine learning, though it is incremental by extending prior work to noisy scenarios.
The paper tackles the problem of non-uniqueness in data-driven discovery of physical laws from noisy data, developing a mathematical framework and algorithms to analyze and detect uniqueness in PDEs, with numerical experiments showing effectiveness.
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.