Compressive-sensing-assisted mixed integer optimization for dynamical system discovery with highly noisy data

TL;DR

This paper introduces a novel CS-MIO method combining compressive sensing and mixed integer optimization to accurately identify governing equations of dynamical systems from highly noisy data, outperforming existing methods.

Contribution

The paper presents a new CS-MIO approach that enables exact discovery of dynamical equations from noisy data using mixed integer optimization with regularization.

Findings

Successfully discovers equations from data with noise up to 100 times higher than previous methods.

Effective in high-dimensional systems like Lorenz 96.

Outperforms state-of-the-art techniques in noisy environments.

Abstract

The identification of governing equations for dynamical systems is everlasting challenges for the fundamental research in science and engineering. Machine learning has exhibited great success to learn and predict dynamical systems from data. However, the fundamental challenges still exist: discovering the exact governing equations from highly noisy data. In present work, we propose a compressive sensing-assisted mixed integer optimization (CS-MIO) method to make a step forward from a modern discrete optimization lens. In particular, we first formulate the problem into a mixed integer optimization model. The discrete optimization nature of the model leads to exact variable selection by means of cardinality constraint, and hereby powerful capability of exact discovery of governing equations from noisy data. Such capability is further enhanced by incorporating compressive sensing and regularization techniques for highly noisy data and high-dimensional problems. The case studies on classical dynamical systems have shown that CS-MIO can discover the exact governing equations from large-noise data, with up to two orders of magnitude larger noise comparing with state-of-the-art method. We also show its effectiveness for high-dimensional dynamical system identification through the chaotic Lorenz 96 system.