Noisy dynamic simulations in the presence of symmetry: data alignment and model reduction

TL;DR

This paper introduces eigenvector-based techniques for aligning noisy trajectory data with symmetries, enabling effective model reduction and low-dimensional parametrization in complex dynamical systems.

Contribution

It presents a novel approach combining symmetry removal and dimensionality reduction using vector diffusion maps, applicable to noisy data from evolution equations with symmetries.

Findings

Successful symmetry quotienting of noisy data in PDE and stochastic simulations

Effective low-dimensional parametrization of complex dynamical systems

Demonstration of combined symmetry removal and dimensionality reduction technique

Abstract

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.