  In this paper, we give an in-depth error analysis for surrogate models
generated by a variant of the Sparse Identification of Nonlinear Dynamics
(SINDy) method. We start with an overview of a variety of non-linear system
identification techniques, namely, SINDy, weak-SINDy, and the occupation kernel
method. Under the assumption that the dynamics are a finite linear combination
of a set of basis functions, these methods establish a matrix equation to
recover coefficients. We illuminate the structural similarities between these
techniques and establish a projection property for the weak-SINDy technique.
Following the overview, we analyze the error of surrogate models generated by a
simplified version of weak-SINDy. In particular, under the assumption of
boundedness of a composition operator given by the solution, we show that (i)
the surrogate dynamics converges towards the true dynamics and (ii) the
solution of the surrogate model is reasonably close to the true solution.
Finally, as an application, we discuss the use of a combination of weak-SINDy
surrogate modeling and proper orthogonal decomposition (POD) to build a
surrogate model for partial differential equations (PDEs).
