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
reliability.
