Compressive-Sensing-Assisted Mixed Integer Optimization for Dynamical System Discovery With Highly Noisy Data

The identification of governing equations for dynamical systems is an everlasting challenge for the fundamental research in science and engineering. Machine learning has exhibited great success in learn and predicting dynamical systems from data. However, the fundamental challenges still exist: discovering the exact governing equations from highly noisy data. In the 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 thereby 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 compared with state-of-the-art methods. We also show its effectiveness for high-dimensional dynamical system identification through the chaotic Lorenz 96 system.