This article overviews how gradient flows, and discretizations thereof, are useful to design and analyze optimization and sampling algorithms. The interplay between optimization, sampling, and gradient flows is an active research area; our goal is to provide an accessible and lively introduction to some core ideas, emphasizing that gradient flows uncover the conceptual unity behind many optimization and sampling algorithms, and that they give a rich mathematical framework for their rigorous analysis.

翻译：本文综述了梯度流及其离散化如何有助于设计和分析优化及采样算法。优化、采样与梯度流之间的相互作用是一个活跃的研究领域；我们的目标是提供一个通俗易懂且生动的核心思想导引，强调梯度流揭示了众多优化与采样算法背后的概念统一性，并为它们的严谨分析提供了丰富的数学框架。