Calculus and geometry are ubiquitous in the theoretical modelling of scientific phenomena, but have historically been very challenging to apply directly to real data as statistics. Diffusion geometry is a new theory that reformulates classical calculus and geometry in terms of a diffusion process, allowing these theories to generalise beyond manifolds and be computed from data. This work introduces a new computational framework for diffusion geometry that substantially broadens its practical scope and improves its precision, robustness to noise, and computational complexity. We present a range of new computational methods, including all the standard objects from vector calculus and Riemannian geometry, and apply them to solve spatial PDEs and vector field flows, find geodesic (intrinsic) distances, curvature, and several new topological tools like de Rham cohomology, circular coordinates, and Morse theory. These methods are data-driven, scalable, and can exploit highly optimised numerical tools for linear algebra.

翻译：微积分与几何学在科学现象的理论建模中无处不在，但历史上一直难以直接作为统计学方法应用于真实数据。扩散几何是一种新理论，它将经典微积分与几何学重新表述为扩散过程，使得这些理论能够推广到流形之外，并能从数据中计算得出。本文提出了一种新的扩散几何计算框架，该框架显著拓宽了其实际应用范围，并提升了其精度、噪声鲁棒性及计算复杂度。我们提出了一系列新的计算方法，涵盖了向量微积分和黎曼几何中的所有标准对象，并将其应用于求解空间偏微分方程和向量场流、寻找测地线（本征）距离、曲率，以及若干新的拓扑工具，如德拉姆上同调、圆坐标和莫尔斯理论。这些方法是数据驱动的、可扩展的，并且能够利用高度优化的线性代数数值工具。