Score-based generative models learn a family of noise-conditional score functions corresponding to the data density perturbed with increasingly large amounts of noise. These perturbed data densities are tied together by the Fokker-Planck equation (FPE), a partial differential equation (PDE) governing the spatial-temporal evolution of a density undergoing a diffusion process. In this work, we derive a corresponding equation, called the score FPE that characterizes the noise-conditional scores of the perturbed data densities (i.e., their gradients). Surprisingly, despite impressive empirical performance, we observe that scores learned via denoising score matching (DSM) do not satisfy the underlying score FPE. We prove that satisfying the FPE is desirable as it improves the likelihood and the degree of conservativity. Hence, we propose to regularize the DSM objective to enforce satisfaction of the score FPE, and we show the effectiveness of this approach across various datasets.

翻译：基于分数的基因化模型学会了一组与数据密度相对应的噪声条件评分函数,其数量越来越大。这些扰动的数据密度由Fokker-Planck方程式(FPE)联系在一起,后者是一个部分差异方程式(PDE),它制约着正在扩散的密度的空间-时空演化过程。在这项工作中,我们得出一个相应的方程式,称为分数FPE,它代表着受扰动数据密度(即其梯度)的噪声条件评分特征。 令人惊讶的是,尽管我们观察到了令人印象深刻的经验性表现,但通过拆分得分比对得分(DSM)的得分并不能满足基本得分的FPE。我们证明满足FPE是可取的,因为它能提高可能性和耐受力的程度。因此,我们建议调整DSM目标,以强制得分的FPE值(即其梯度)的满意度。我们在各个数据集展示了这一方法的有效性。