The Right Space for Dynamics: Numerics with Diffeomorphism Equivariance

Abstract

Among many (equivalent, via invertible transformations) representations of the evolution of a dynamical system, which one is to be preferred? Here we show how the use of infinite-dimensional Lie group theory (and its numerical implementation) allows us to single out one representation, by selecting an element of the group of diffeomorphisms acting on the dynamical system. We present and discuss several types of ``phase conditions" defining the selected representation, and illustrate their computational implementation. Study of dynamics modulo diffeomorphisms ``liberates" mathematical modeling of physical phenomena from a user's preferred coordinates, and spontaneously selects a ``right latent space" for the system.