Equation discovery from data: promise and pitfalls, from rabbits to Mars

Abstract

The problem of equation discovery seeks to reconstruct the underlying dynamics of a time-varying system from observations of the system, and moreover to do so in an instructive way such that we may understand these underlying dynamics from the reconstruction.
This article illustrates one type of modern equation discovery method (sparse identification of nonlinear dynamics, or SINDy) in the context of two classic problems. The presentation is in a tutorial style intended to be accessible to students, and could form a useful module in undergraduate or graduate courses in modelling, data analysis, or numerical methods. In this style we explore the strengths and limitations of these methods. We also demonstrate, through use of a carefully constructed example, a new result about the relationship between the reconstructed and true models when a na\"ive polynomial basis is used.