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Symbolic recovery of differential equations is the ambitious attempt atautomating the derivation of governing equations with the use of machinelearning techniques. In contrast to classical methods which assume thestructure of the equation to be known and focus on the estimation of specificparameters, these algorithms aim to learn the structure and the parameterssimultaneously. While the uniqueness and, therefore, the identifiability ofparameters of governing equations are a well-addressed problem in the field ofparameter estimation, it has not been investigated for symbolic recovery.However, this problem should be even more present in this field since thealgorithms aim to cover larger spaces of governing equations. In this paper, weinvestigate under which conditions a solution of a differential equation doesnot uniquely determine the equation itself. For various classes of differentialequations, we provide both necessary and sufficient conditions for a functionto uniquely determine the corresponding differential equation. We then use ourresults to devise numerical algorithms aiming to determine whether a functionsolves a differential equation uniquely. Finally, we provide extensivenumerical experiments showing that our algorithms can indeed guarantee theuniqueness of the learned governing differential equation, without assuming anyknowledge about the analytic form of function, thereby ensuring the reliabilityof the learned equation.

Symbolic Recovery of Differential Equations: The Identifiability Problem