  Modern societies have an abundance of data yet good system models are rare.
Unfortunately, many of the current system identification and machine learning
techniques fail to generalize outside of the training set, producing models
that violate basic physical laws. This work proposes a novel method for the
Sparse Identification of Nonlinear Dynamics with Side Information (SINDy-SI).
SINDy-SI is an iterative method that uses Sum-of-Squares (SOS) programming to
learn optimally fitted models while guaranteeing that the learned model
satisfies side information, such as symmetry's and physical laws. Guided by the
principle of Occam's razor, that the simplest or most regularized best fitted
model is typically the superior choice, during each iteration SINDy-SI prunes
the basis functions associated with small coefficients, yielding a sparse
dynamical model upon termination. Through several numerical experiments we will
show how the combination of side information constraints and sparse polynomial
representation cultivates dynamical models that obey known physical laws while
displaying impressive generalized performance beyond the training set.
