发表机构
Kerala School of Mathematics（喀拉拉数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在非标准分析框架下构建了基于得分的生成建模的超有限框架，推导了相关恒等式、公式并分析了二阶一致性，建立了离散网格动力学与生成建模核心环节的关联。
AI 中文摘要
基于得分的扩散模型通常采用连续时间随机微分方程和测度论随机微积分构建。本文在非标准分析框架内构建了基于得分的生成建模的超有限形式化：从超有限网格上的内部扩散过程出发，推导其无穷小生成元并建立与经典福克-普朗克方程的对应关系；随后得到超有限后向均值恒等式，该恒等式可产生逆时漂移并对逆时随机微分方程进行构造性推导。基于上述结果，本文证明最小化内部得分匹配目标可恢复逆时动力学所需的得分函数，从而在超有限层面直接将得分估计与生成采样关联起来。在适当假设下，本文进一步推导超有限吉萨诺夫公式，并建立似然优化与费希尔散度目标之间的关系。最后，本文分析超有限动力学的二阶一致性，证明主导修正项显式依赖于增量分布的四阶矩，其中高斯值κ=3可消除主导色散贡献。综上，这些结果为基于扩散的生成建模提供了统一的超有限框架，同时为进一步扩展奠定基础，该框架在统一的非标准设定内连接了离散网格动力学、逆时扩散、得分匹配和基于似然的形式化。
英文摘要
Score-based diffusion models are typically formulated using continuous-time stochastic differential equations and measure-theoretic stochastic calculus. In this paper, we develop a hyperfinite formulation of score-based generative modeling within the framework of Nonstandard Analysis. Starting from an internal diffusion process on a hyperfinite grid, we derive the associated infinitesimal generator and establish its correspondence with the classical Fokker--Planck equation. We then obtain a hyperfinite backward-mean identity that yields the reverse-time drift and provides a constructive derivation of the reverse-time SDE. Building on these results, we show that minimization of an internal score-matching objective recovers the score function required by the reverse-time dynamics, thereby connecting score estimation with generative sampling directly at the hyperfinite level. Under suitable assumptions, we further derive a hyperfinite Girsanov formula and establish a relationship between likelihood optimization and Fisher-divergence objectives. Finally, we analyze the second-order consistency of the hyperfinite dynamics and show that the leading correction term depends explicitly on the fourth moment of the increment distribution, with the Gaussian value $κ=3$ eliminating the leading dispersion contribution. Taken together, these results provide a unified hyperfinite framework for diffusion-based generative modeling--while laying foundations for further extensions--that links discrete grid dynamics, reverse-time diffusion, score matching, and likelihood-based formulations within a common nonstandard setting.
