# Symbolic Recovery of Differential Equations: The Identifiability Problem

arXiv: https://arxiv.org/abs/2210.08342 · HTML: https://mlsanity.com/paper/2210.08342/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 74 references that are in the map.
- First posted on arXiv 15 Oct 2022.

## Builds on (most cited first, 17 of 17)

- [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)
- [Contemporary Symbolic Regression Methods and their Relative Performance](https://mlsanity.com/paper/2107.14351/) (2021)
- [End-to-end symbolic regression with transformers](https://mlsanity.com/paper/2204.10532/) (2022)
- [Neural Symbolic Regression that Scales](https://mlsanity.com/paper/2106.06427/) (2021)
- [Physics-informed learning of governing equations from scarce data](https://mlsanity.com/paper/2005.03448/) (2020)
- [Learning Equations for Extrapolation and Control](https://mlsanity.com/paper/1806.07259/) (2018)
- [SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamics](https://mlsanity.com/paper/2004.02322/) (2020)
- [Extrapolation and learning equations](https://mlsanity.com/paper/1610.02995/) (2016)
- [DeepMoD: Deep learning for Model Discovery in noisy data](https://mlsanity.com/paper/1904.09406/) (2019)
- [Symbolic Physics Learner: Discovering governing equations via Monte Carlo tree search](https://mlsanity.com/paper/2205.13134/) (2022)
- [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 outside the Black-Box: The pursuit of interpretable models](https://mlsanity.com/paper/2011.08596/) (2020)
- [Learning Partial Differential Equations from Data Using Neural Networks](https://mlsanity.com/paper/1910.10262/) (2019)
- [D-CIPHER: Discovery of Closed-form Partial Differential Equations](https://mlsanity.com/paper/2206.10586/) (2022)
- [Global Identifiability of Differential Models](https://mlsanity.com/paper/1801.08112/) (2018)
- [Computing all identifiable functions of parameters for ODE models](https://mlsanity.com/paper/2004.07774/) (2020)

## Built on by (most cited first, 8 of 8)

- [Identifiability Challenges in Sparse Linear Ordinary Differential Equations](https://mlsanity.com/paper/2506.09816/) (2025)
- [Grammar-based Ordinary Differential Equation Discovery](https://mlsanity.com/paper/2504.02630/) (2025)
- [When is a System Discoverable from Data? Discovery Requires Chaos](https://mlsanity.com/paper/2511.08860/) (2025)
- [Data-efficient Kernel Methods for Learning Hamiltonian Systems](https://mlsanity.com/paper/2509.17154/) (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)
- [Deflation-PINNs: Learning Multiple Solutions for PDEs and Landau-de Gennes](https://mlsanity.com/paper/2603.27936/) (2026)
- [Identifying ODEs from Unstructured Data with Causal Representation Learning](https://mlsanity.com/paper/2609.37083/) (2026)

## Abstract (arXiv)

Symbolic recovery of differential equations is the ambitious attempt at automating the derivation of governing equations with the use of machine learning techniques. In contrast to classical methods which assume the structure of the equation to be known and focus on the estimation of specific parameters, these algorithms aim to learn the structure and the parameters simultaneously. While the uniqueness and, therefore, the identifiability of parameters of governing equations are a well-addressed problem in the field of parameter estimation, it has not been investigated for symbolic recovery. However, this problem should be even more present in this field since the algorithms aim to cover larger spaces of governing equations. In this paper, we investigate under which conditions a solution of a differential equation does not uniquely determine the equation itself. For various classes of differential equations, we provide both necessary and sufficient conditions for a function to uniquely determine the corresponding differential equation. We then use our results to devise numerical algorithms aiming to determine whether a function solves a differential equation uniquely. Finally, we provide extensive numerical experiments showing that our algorithms can indeed guarantee the uniqueness of the learned governing differential equation, without assuming any knowledge about the analytic form of function, thereby ensuring the reliability of the learned equation.