  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 first analyse existing methods for computing length-minimising
geodesics. We find that these are not suitable for obtaining valid paths, and
thus, geodesic distances. We remedy these shortcomings by leveraging numerical
tools from differential geometry, which provide the means to obtain
Hamiltonian-conserving geodesics. Second, we propose a model-based
parameterisation for distance fields and geodesic flows on continuous
manifolds. Our approach exploits a manifold-aware extension to the Eikonal
equation, eliminating the need for approximations or discretisation. Finally,
we develop a curvature-based training mechanism, sampling and scaling points in
regions of the manifold exhibiting larger values of the Ricci scalar. This
sampling and scaling approach ensures that we capture regions of the manifold
subject to higher degrees of geodesic deviation. Our proposed methods provide
principled means to compute valid geodesics and geodesic distances on
manifolds. This work opens opportunities for latent-space interpolation,
optimal control, and distance computation on differentiable manifolds.
