  Accurately modeling the nonlinear dynamics of a system from measurement data
is a challenging yet vital topic. The sparse identification of nonlinear
dynamics (SINDy) algorithm is one approach to discover dynamical systems models
from data. Although extensions have been developed to identify implicit
dynamics, or dynamics described by rational functions, these extensions are
extremely sensitive to noise. In this work, we develop SINDy-PI (parallel,
implicit), a robust variant of the SINDy algorithm to identify implicit
dynamics and rational nonlinearities. The SINDy-PI framework includes multiple
optimization algorithms and a principled approach to model selection. We
demonstrate the ability of this algorithm to learn implicit ordinary and
partial differential equations and conservation laws from limited and noisy
data. In particular, we show that the proposed approach is several orders of
magnitude more noise robust than previous approaches, and may be used to
identify a class of complex ODE and PDE dynamics that were previously
unattainable with SINDy, including for the double pendulum dynamics and the
Belousov Zhabotinsky (BZ) reaction.
