Reconstruction of Ordinary Differential Equations From Time Series Data

TL;DR

This paper introduces a numerical method for reconstructing ordinary differential equations from time series data using sparse basis learning, improving accuracy over traditional least-squares methods, and validates it on systems with fixed points, oscillations, and chaos.

Contribution

The paper presents a novel sparse reconstruction technique for ODEs from data, validated on complex systems, enabling identification of unknown dynamical models.

Findings

Sparse representations yield more accurate ODE reconstructions.

The method accurately reconstructs trajectories for systems with fixed points and oscillations.

Reconstructed models capture key dynamical features like Lyapunov exponents.

Abstract

We develop a numerical method to reconstruct systems of ordinary differential equations (ODEs) from time series data without {\it a priori} knowledge of the underlying ODEs using sparse basis learning and sparse function reconstruction. We show that employing sparse representations provides more accurate ODE reconstruction compared to least-squares reconstruction techniques for a given amount of time series data. We test and validate the ODE reconstruction method on known 1D, 2D, and 3D systems of ODEs. The 1D system possesses two stable fixed points; the 2D system possesses an oscillatory fixed point with closed orbits; and the 3D system displays chaotic dynamics on a strange attractor. We determine the amount of data required to achieve an error in the reconstructed functions to less than $0.1\%$. For the reconstructed 1D and 2D systems, we are able to match the trajectories from the original ODEs even at long times. For the 3D system with chaotic dynamics, as expected, the trajectories from the original and reconstructed systems do not match at long times, but the reconstructed and original models possess similar Lyapunov exponents. Now that we have validated this ODE reconstruction method on known models, it can be employed in future studies to identify new systems of ODEs using time series data from deterministic systems for which there is no currently known ODE model.