Numerical aspects for approximating governing equations using data

Highlights: • Proposed the use of a large number of short trajectory data and discuss sampling strategies. • Presented several effective numerical algorithms for obtaining the equation approximation. • Presented an error estimate on the solution obtained by the numerically generated governing equations. • Provided an extensive set of examples to demonstrate the effectiveness of the methods. -- Abstract: We present effective numerical algorithms for approximating unknown governing differential equations from measurement data. We employ a set of standard basis functions, e.g., polynomials, to approximate the governing equation with high accuracy. Upon recasting the problem into a function approximation problem, we discuss several important aspects for accurate approximation. Most notably, we discuss the importance of using a large number of short bursts of trajectory data, rather than using data from a single long trajectory. Several options for the numerical algorithms to perform accurate approximation are then presented, along with an error estimate of the final equation approximation. We then present an extensive set of numerical examples of both linear and nonlinear systems to demonstrate the properties and effectiveness of our equation approximation algorithms.