Symbolic Recovery of Differential Equations: The Identifiability Problem

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.