Automatically identifying ordinary differential equations from data

TL;DR

This paper presents a method for automatically discovering nonlinear ordinary differential equations from empirical data by combining denoising, sparse regression, and uncertainty quantification, enabling accurate system identification.

Contribution

The authors introduce a novel integrated approach that combines denoising, sparse regression, and bootstrap confidence intervals for automatic differential equation discovery from noisy data.

Findings

Successfully identifies 3D systems with moderate data and high signal quality.

Consistently performs well across various initial conditions and noise levels.

Potential to accelerate understanding of complex systems in data-rich fields.

Abstract

Discovering nonlinear differential equations that describe system dynamics from empirical data is a fundamental challenge in contemporary science. Here, we propose a methodology to identify dynamical laws by integrating denoising techniques to smooth the signal, sparse regression to identify the relevant parameters, and bootstrap confidence intervals to quantify the uncertainty of the estimates. We evaluate our method on well-known ordinary differential equations with an ensemble of random initial conditions, time series of increasing length, and varying signal-to-noise ratios. Our algorithm consistently identifies three-dimensional systems, given moderately-sized time series and high levels of signal quality relative to background noise. By accurately discovering dynamical systems automatically, our methodology has the potential to impact the understanding of complex systems, especially in fields where data are abundant, but developing mathematical models demands considerable effort.