Score-based diffusion generative models have recently emerged as a powerful tool for modelling complex data distributions. These models aim at learning the score function, which defines a map from a known probability distribution to the target data distribution via deterministic or stochastic differential equations (SDEs). The score function is typically estimated from data using a variety of approximation techniques, such as denoising or sliced score matching, Hyvärien's method, or Schrödinger bridges. In this paper, we derive an exact, closed-form, expression for the score function for a broad class of nonlinear diffusion generative models. Our approach combines modern stochastic analysis tools such as Malliavin derivatives and their adjoint operators (Skorokhod integrals or Malliavin Divergence) with a new Bismut-type formula. The resulting expression for the score function can be written entirely in terms of the first and second variation processes, with all Malliavin derivatives systematically eliminated, thereby enhancing its practical applicability. The theoretical framework presented in this work offers a principled foundation for advancing score estimation methods in generative modelling, enabling the design of new sampling algorithms for complex probability distributions. Our results can be extended to broader classes of stochastic differential equations, opening new directions for the development of score-based diffusion generative models.

翻译：基于得分的扩散生成模型近年来已成为建模复杂数据分布的有力工具。这些模型旨在学习得分函数，该函数通过确定性或随机微分方程（SDEs）定义了从已知概率分布到目标数据分布的映射。得分函数通常使用多种近似技术从数据中估计，例如去噪或切片得分匹配、Hyvärien方法或薛定谔桥。在本文中，我们针对一大类非线性扩散生成模型推导出了得分函数的精确闭式表达式。我们的方法将现代随机分析工具（如Malliavin导数及其伴随算子（Skorokhod积分或Malliavin散度））与新的Bismut型公式相结合。所得得分函数表达式可完全用一阶和二阶变分过程表示，所有Malliavin导数被系统性地消除，从而增强了其实际适用性。本文提出的理论框架为推进生成建模中的得分估计方法提供了原则性基础，使得能够为复杂概率分布设计新的采样算法。我们的结果可扩展至更广泛的随机微分方程类别，为基于得分的扩散生成模型的发展开辟了新方向。