Numerical Aspects for Approximating Governing Equations Using Data

TL;DR

This paper introduces numerical algorithms for accurately recovering unknown differential equations from measurement data, emphasizing the use of multiple short data bursts and polynomial basis functions for improved approximation.

Contribution

It proposes a novel approach that leverages multiple short data segments and standard basis functions to enhance the accuracy of differential equation recovery from data.

Findings

Effective algorithms for local differential equation recovery

Use of multiple short data bursts improves accuracy

Numerical examples demonstrate robustness for linear and nonlinear systems

Abstract

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.