Data-driven discovery of free-form governing differential equations

Abstract

We present a method of discovering governing differential equations from data without the need to specify a priori the terms to appear in the equation. The input to our method is a dataset (or ensemble of datasets) corresponding to a particular solution (or ensemble of particular solutions) of a differential equation. The output is a human-readable differential equation with parameters calibrated to the individual particular solutions provided. The key to our method is to learn differentiable models of the data that subsequently serve as inputs to a genetic programming algorithm in which graphs specify computation over arbitrary compositions of functions, parameters, and (potentially differential) operators on functions. Differential operators are composed and evaluated using recursive application of automatic differentiation, allowing our algorithm to explore arbitrary compositions of operators without the need for human intervention. We also demonstrate an active learning process to identify and remedy deficiencies in the proposed governing equations.