计算扩散几何
Computing Diffusion Geometry
AI总结：
本文提出了一种新的计算框架，用于扩散几何，扩展了其应用范围，提高了精度和鲁棒性，并引入了多种新的拓扑工具和计算方法。
摘要：
微积分和几何学在科学现象的理论建模中无处不在，但历史上直接应用于真实数据作为统计学却非常具有挑战性。扩散几何是一种新的理论，它用扩散过程重新表述经典微积分和几何学，使这些理论能够超越流形并从数据中计算出来。本文介绍了一种新的计算框架，大大扩展了扩散几何的实用范围，并提高了其精度、对噪声的鲁棒性和计算复杂性。我们提出了一系列新的计算方法，包括所有标准的向量微积分和黎曼几何对象，并将其应用于解决空间偏微分方程和向量场流动、寻找测地线（内在）距离、曲率以及几种新的拓扑工具，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，并且可以利用高度优化的线性代数数值工具。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
