  We process snapshots of trajectories of evolution equations with intrinsic
symmetries, and demonstrate the use of recently developed eigenvector-based
techniques to successfully quotient out the degrees of freedom associated with
the symmetries in the presence of noise. Our illustrative examples include a
one-dimensional evolutionary partial differential (the Kuramoto-Sivashinsky)
equation with periodic boundary conditions, as well as a stochastic simulation
of nematic liquid crystals which can be effectively modeled through a nonlinear
Smoluchowski equation on the surface of a sphere. This is a useful first step
towards data mining the "symmetry-adjusted" ensemble of snapshots in search of
an accurate low-dimensional parametrization (and the associated reduction of
the original dynamical system). We also demonstrate a technique ("vector
diffusion maps") that combines, in a single formulation, the symmetry removal
step and the dimensionality reduction step.

Computational Physics