想搞懂扩散模型、得分匹配背后的数学原理？这篇论文从变分角度讲清楚了SDE和ELBO，适合想深入理论的朋友。
这篇论文从变分视角系统介绍了随机微分方程（SDE）在生成机器学习中的应用。它推导了证据下界（ELBO）并作为讨论扩散模型、得分匹配和流匹配的统一框架。论文使用一维密度建模问题比较了不同参数化的效果。
Introduction to Stochastic Differential Equations for Generative Machine Learning: A Variational Perspective
The use of ordinary and stochastic differential equations has led to substantial progress in generative machine learning with applications to, for example, image, video and biomolecule generation. This paper provides a self-contained and informal introduction to the differential equations, the probabilistic framework for using them in generative modeling and the Fokker--Planck equation that governs the temporal evolution of the marginal distribution of the stochastic variables of the differential equations. The variational lower bound on the log-likelihood (the evidence lower bound, ELBO) is derived and used as a general starting point for a discussion of diffusion models, score matching, and flow matching. All of these approaches may be viewed as specific parameterizations of the most general variational approach. A one-dimensional density modeling problem is used as a simple example to compare different parameterizations.