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 that a physical law is 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 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.
翻译:物理法学习是用机器学习技术使治理方程式的衍生自动化的模棱两可的尝试。 但是,目前的文献只侧重于制定实现这一目标的方法,目前还缺少一个理论基础。因此,本文件应作为第一步,为学习物理法建立一个全面的理论框架,目的是为算法提供可靠性。一个关键问题是,治理方程式可能不是由给定数据所决定的独特方程式。我们将研究在通常情况下物理法是由普通或部分差异方程式描述的问题。对于不同类别的差异方程式,我们为一种功能提供了必要和充分的条件,以便独一地确定管理这一现象的不同方程式。我们然后利用我们的结果来设计数字方程式以确定函数是否解决差异方程式的独特性。最后,我们提供了广泛的数字实验,表明我们的算法与学习物理法的共同方法相结合,确实能够保证学习独特的治理方程式,而不必假定对功能的任何了解,从而确保可靠性。