Physical law learning is the ambiguous attempt at automating the derivation
    of governing equations with the use of machine learning techniques. The current
    literature focuses however solely on the development of methods to achieve this
    goal, and a theoretical foundation is at present missing. This paper shall thus
    serve as a first step to build a comprehensive theoretical framework for
    learning physical laws, aiming to provide reliability to according algorithms.
    One key problem consists in the fact that the governing equations might not be
    uniquely determined by the given data. We will study this problem in the common
    situation of having a physical law be described by an ordinary or partial
    differential equation. For various different classes of differential equations,
    we provide both necessary and sufficient conditions for a function from a given
    function class to uniquely determine the differential equation which is
    governing the phenomenon. We then use our results to devise numerical
    algorithms to determine whether a function solves a differential equation
    uniquely. Finally, we provide extensive numerical experiments showing that our
    algorithms in combination with common approaches for learning physical laws
    indeed allow to guarantee that a unique governing differential equation is
    learnt, without assuming any knowledge about the function, thereby ensuring

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