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Numerical aspects for approximating governing equations using data. (English) Zbl 1451.65008
Summary: 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.

MSC:
65C20 Probabilistic models, generic numerical methods in probability and statistics
37M10 Time series analysis of dynamical systems
34A55 Inverse problems involving ordinary differential equations
34A09 Implicit ordinary differential equations, differential-algebraic equations
62G08 Nonparametric regression and quantile regression
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