  In recent years there has been a push to discover the governing equations
dynamical systems directly from measurements of the state, often motivated by
systems that are too complex to directly model. Although there has been
substantial work put into such a discovery, doing so in the case of large noise
has proved challenging. Here we develop an algorithm for Simultaneous
Identification and Denoising of a Dynamical System (SIDDS). We infer the noise
in the state measurements by requiring that the denoised data satisfies the
dynamical system with an equality constraint. This is unlike existing work
where the mismatch in the dynamical system is treated as a penalty in the
objective. We assume the dynamics is represented in a pre-defined basis and
develop a sequential quadratic programming approach to solve the SIDDS problem
featuring a direct solution of KKT system with a specialized preconditioner. In
addition, we show how we can include sparsity promoting regularization using an
iteratively reweighted least squares approach. The resulting algorithm leads to
estimates of the dynamical system that approximately achieve the Cram\'er-Rao
lower bound and, with sparsity promotion, can correctly identify the sparsity
structure for higher levels of noise than existing techniques. Moreover,
because SIDDS decouples the data from the evolution of the dynamical system, we
show how to modify the problem to accurately identify systems from low sample
rate measurements. The inverse problem approach and solution framework used by
SIDDS has the potential to be expanded to related problems identifying
governing equations from noisy data.
