# Robust identifiability for symbolic recovery of differential equations

arXiv: https://arxiv.org/abs/2410.09938 · HTML: https://mlsanity.com/paper/2410.09938/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 24 references that are in the map.
- First posted on arXiv 13 Oct 2024.

## Builds on (most cited first, 6 of 6)

- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients](https://mlsanity.com/paper/1912.04871/) (2019)
- [Physics-informed learning of governing equations from scarce data](https://mlsanity.com/paper/2005.03448/) (2020)
- [SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamics](https://mlsanity.com/paper/2004.02322/) (2020)
- [DLGA-PDE: Discovery of PDEs with incomplete candidate library via combination of deep learning and genetic algorithm](https://mlsanity.com/paper/2001.07305/) (2020)
- [Learning Partial Differential Equations from Data Using Neural Networks](https://mlsanity.com/paper/1910.10262/) (2019)

## Built on by (most cited first, 3 of 3)

- [When is a System Discoverable from Data? Discovery Requires Chaos](https://mlsanity.com/paper/2511.08860/) (2025)
- [Physically consistent model learning for reaction-diffusion systems](https://mlsanity.com/paper/2512.14240/) (2025)
- [Symbolic recovery of PDEs from measurement data](https://mlsanity.com/paper/2602.15603/) (2026)

## Abstract (arXiv)

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.