Sparse Identification of Nonlinear Dynamical Systems via Reweighted $\ell_1$-regularized Least Squares

The structure of these differential equations is usually determined by observing the system and inferring relationships between variables, or derived from fundamental axioms and mathematical reasoning. Examples of the empirical method include Johannes Kepler and Isaac Newton's approaches in deriving laws of planetary motion. The accurate measurements of planet trajectories by Tycho Brahe enabled Kepler to empirically determine the laws that govern the motion of elliptic orbits. Newton, in turn, was able to derive the law of universal gravitation by inductive reasoning. Solving models derived from fundamental laws, either analytically or numerically, has proven to be a useful approach in engineering to produce reliable systems. However, the derived models often rely on simplifying assumptions that may not explain complex phenomena, leading to a mismatch between predictions and observations. Moreover, parameters of these models may need to be estimated indirectly from system observables.