  Sparse system identification is the data-driven process of obtaining
parsimonious differential equations that describe the evolution of a dynamical
system, balancing model complexity and accuracy. There has been rapid
innovation in system identification across scientific domains, but there
remains a gap in the literature for large-scale methodological comparisons that
are evaluated on a variety of dynamical systems. In this work, we
systematically benchmark sparse regression variants by utilizing the dysts
standardized database of chaotic systems. In particular, we demonstrate how
this open-source tool can be used to quantitatively compare different methods
of system identification. To illustrate how this benchmark can be utilized, we
perform a large comparison of four algorithms for solving the sparse
identification of nonlinear dynamics (SINDy) optimization problem, finding
strong performance of the original algorithm and a recent mixed-integer
discrete algorithm. In all cases, we used ensembling to improve the noise
robustness of SINDy and provide statistical comparisons. In addition, we show
very compelling evidence that the weak SINDy formulation provides significant
improvements over the traditional method, even on clean data. Lastly, we
investigate how Pareto-optimal models generated from SINDy algorithms depend on
the properties of the equations, finding that the performance shows no
significant dependence on a set of dynamical properties that quantify the
amount of chaos, scale separation, degree of nonlinearity, and the syntactic
complexity.
