Numerical Aspects for Approximating Governing Equations Using Data
This work addresses the challenge of data-driven equation discovery for researchers in computational science, but it appears incremental as it focuses on numerical aspects and error estimates without introducing a fundamentally new method.
The paper tackles the problem of recovering unknown governing differential equations from measurement data by presenting numerical algorithms that use standard basis functions for approximation, and demonstrates their effectiveness through extensive examples of linear and nonlinear systems.
We present effective numerical algorithms for locally recovering 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 recovery algorithms.