Manifold-augmented Eikonal Equations: Geodesic Distances and Flows on Differentiable Manifolds
This work addresses the need for efficient geodesic computations in reduced-order modeling and statistical inference on manifolds, though it appears incremental as it builds on existing Eikonal equation methods.
The authors tackled the problem of computing geodesic distances and flows on differentiable manifolds discovered by machine learning models, proposing a model-based parameterization using a manifold-augmented Eikonal equation, which enables obtaining globally length-minimizing curves directly.
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.