Discovering and modelling dynamics on latent manifolds with neural geodesic flows

Julian Bürge, Lewis O'Donnell, Ben Moseley

Keywords: Riemannian manifold, dynamical systems, geometric deep learning, geodesic solver, generative modelling

TL;DR: We present neural geodesic flows, a framework that learns both a latent Riemannian metric and coordinate transforms to model dynamical systems evolving along geodesics.

Abstract: We present neural geodesic flows (NGFs), a framework for discovering and modelling dynamical systems which assumes the system evolves along the geodesics of a latent Riemannian manifold. Both the metric of the manifold and coordinate transforms between observational data and the manifold are simultaneously learned from observational data. Whilst most approaches for dynamical system modelling make rigid physical assumptions, NGFs only assume geometrical properties of the system's evolution and have the capacity to model a wide range of systems. NGFs are trained in an end-to-end fashion, backpropagating through the coordinate transforms, a numerical geodesic solver on the manifold, and the metric of the manifold, and several techniques such as residual learning and extreme value soft-clipping are required to ensure stable gradient flow. We show that NGFs can accurately model particle flow on a sphere and the two-body problem, and can be applied to generative modelling on arbitrary manifolds; with further work, NGFs could provide a powerful geometry-based framework for dynamical systems modelling.

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Submission Number: 40