  Sparse regression has emerged as a popular technique for learning dynamical
systems from temporal data, beginning with the SINDy (Sparse Identification of
Nonlinear Dynamics) framework proposed by arXiv:1509.03580. Quantifying the
uncertainty inherent in differential equations learned from data remains an
open problem, thus we propose leveraging recent advances in statistical
inference for sparse regression to address this issue. Focusing on systems of
ordinary differential equations (ODEs), SINDy assumes that each equation is a
parsimonious linear combination of a few candidate functions, such as
polynomials, and uses methods such as sequentially-thresholded least squares or
the Lasso to identify a small subset of these functions that govern the
system's dynamics. We instead employ bias-corrected versions of the Lasso and
ridge regression estimators, as well as an empirical Bayes variable selection
technique known as SEMMS, to estimate each ODE as a linear combination of terms
that are statistically significant. We demonstrate through simulations that
this approach allows us to recover the functional terms that correctly describe
the dynamics more often than existing methods that do not account for
uncertainty.
