  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.
