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.

翻译：机器学习模型所发现的流形提供了对底层数据的紧凑表示。这些流形上的测地线定义了局部长度最小化曲线，并提供了距离的概念，这对于降阶建模、统计推断和插值至关重要。在本文中，我们首先分析了现有计算长度最小化测地线的方法。我们发现这些方法不适合获得有效路径，进而无法得到测地距离。我们通过利用微分几何中的数值工具来弥补这些不足，从而提供了获得哈密顿守恒测地线的手段。其次，我们提出了一种基于模型的参数化方法，用于连续流形上的距离场和测地线流。我们的方法利用了对程函方程的流形感知扩展，从而消除了近似或离散化的需要。最后，我们开发了一种基于曲率的训练机制，在流形上具有较大里奇标量值的区域对点进行采样和缩放。这种采样和缩放方法确保我们捕捉到流形上遭受更高程度测地线偏差的区域。我们提出的方法提供了在流形上计算有效测地线和测地距离的原则性手段。这项工作为可微流形上的潜在空间插值、最优控制和距离计算开辟了机遇。