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：基于分数的生成模型学习一系列噪声条件分数函数，这些函数对应于被逐渐增大噪声扰动后的数据密度。这些扰动后的数据密度通过福克-普朗克方程（FPE）相互关联，该方程是描述扩散过程中密度空间-时间演化的偏微分方程。在本工作中，我们推导出一个相应的方程，称为分数FPE，它刻画了扰动数据密度的噪声条件分数（即其梯度）。令人惊讶的是，尽管经验表现优异，我们发现通过去噪分数匹配（DSM）学得的分数并不满足底层分数FPE。我们证明满足FPE是可取的，因为它能提升似然性和保守性程度。因此，我们提出对DSM目标进行正则化处理，以强制满足分数FPE，并在多个数据集上展示了该方法的有效性。