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.

翻译：机器学习模型发现的流形为底层数据提供了紧凑表示。这些流形上的测地线定义了局部长度最小化曲线，并提供了距离概念，这对于降阶建模、统计推断和插值至关重要。本文提出一种基于模型的参数化方法，用于流形上的距离场与测地流，通过求解流形增强程函方程实现。我们展示了流形几何如何影响距离场，并直接利用测地流获取全局长度最小化曲线。该工作为可微流形上的统计学与降阶建模开辟了新机遇。