Manifold-augmented Eikonal Equations: Geodesic Distances and Flows on Differentiable Manifolds

TL;DR

This paper introduces a model-based approach to compute geodesic distances and flows on differentiable manifolds learned by machine learning, enabling improved statistical analysis and reduced-order modeling.

Contribution

It proposes a novel parameterisation using manifold-augmented Eikonal equations to directly obtain geodesic flows and distances on learned manifolds.

Findings

Demonstrates how manifold geometry influences distance fields.

Shows how geodesic flows can be used to find globally length-minimising curves.

Opens new avenues for statistical inference on learned manifolds.

Abstract

Manifolds discovered by machine learning models provide a compact representation of the underlying data. Geodesics on these manifolds define locally length-minimising curves and provide a notion of distance, which are key for reduced-order modelling, statistical inference, and interpolation. In this work, we propose a model-based parameterisation for distance fields and geodesic flows on manifolds, exploiting solutions of a manifold-augmented Eikonal equation. We demonstrate how the geometry of the manifold impacts the distance field, and exploit the geodesic flow to obtain globally length-minimising curves directly. This work opens opportunities for statistics and reduced-order modelling on differentiable manifolds.