  This work is concerned with uncertainty quantification in reduced-order
dynamical system identification. Reduced-order models for system dynamics are
ubiquitous in design and control applications and recent efforts focus on their
data-driven construction. Our starting point is the sparse-identification of
nonlinear dynamics (SINDy) framework, which reformulates system identification
as a regression problem, where unknown functions are approximated from a sparse
subset of an underlying library. In this manuscript, we formulate this system
identification method in a Bayesian framework to handle parameter and
structural model uncertainties. We present a general approach to enforce
sparsity, which builds on the recently introduced class of neuronized priors.
We perform comparisons between different variants such as Lasso, horseshoe, and
spike and slab priors, which are all obtained by modifying a single activation
function. We also outline how state observation noise can be incorporated with
a probabilistic state-space model. The resulting Bayesian regression framework
is robust and simple to implement. We apply the method to two generic numerical
applications, the pendulum and the Lorenz system, and one aerodynamic
application using experimental measurements.
