AI Descartes: Combining Data and Theory for Derivable Scientific Discovery

Scientists have long aimed to discover meaningful formulae which accurately describe experimental data. A common approach is to manually create mathematical models of natural phenomena using domain knowledge, and then fit these models to data. In contrast, machine-learning algorithms automate the construction of accurate data-driven models while consuming large amounts of data. The problem of enforcing logic constraints on the functional form of a learned model (e.g., nonnegativity) has been explored in the literature; however, finding models that are consistent with general background knowledge is an open problem. We develop a method for combining logical reasoning with symbolic regression, enabling principled derivations of models of natural phenomena. We demonstrate these concepts for Kepler's third law of planetary motion, Einstein's relativistic time-dilation law, and Langmuir's theory of adsorption, automatically connecting experimental data with background theory in each case. We show that laws can be discovered from few data points when using formal logical reasoning to distinguish the correct formula from a set of plausible formulas that have similar error on the data. The combination of reasoning with machine learning provides generalizeable insights into key aspects of natural phenomena. We envision that this combination will enable derivable discovery of fundamental laws of science and believe that our work is a crucial first step towards automating the scientific method.