FP-Diffusion：通过强制执行底层得分福克-普朗克方程改进基于得分的扩散模型
FP-Diffusion: Improving Score-based Diffusion Models by Enforcing the Underlying Score Fokker-Planck Equation
AI总结：
本研究发现基于去噪得分匹配的扩散模型学到的得分不满足得分福克-普朗克方程，提出对目标函数正则化以强制满足该方程，提升了模型的似然与保守性。
摘要：
基于得分的生成模型（SGMs）学习一族噪声条件得分函数，这些函数对应被不断增大的噪声扰动的数据密度。这些被扰动的数据密度通过福克-普朗克方程（FPE）相互关联，FPE是一种控制经历扩散过程的密度的时空演化的偏微分方程（PDE）。在本研究中，我们推导了一个对应的方程，称为得分FPE，它刻画了被扰动数据密度的噪声条件得分（即其梯度）。令人意外的是，尽管实证表现出色，我们观察到通过去噪得分匹配（DSM）学习到的得分无法满足底层的得分FPE，而这一性质是真实得分固有的自洽属性。我们证明，满足得分FPE是有益的，因为它能提升似然值和保守性程度。因此，我们提出对DSM目标函数进行正则化，以强制满足得分FPE，并在多个数据集上验证了该方法的有效性。
英文摘要：
Score-based generative models (SGMs) 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 linked 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 the impressive empirical performance, we observe that scores learned through denoising score matching (DSM) fail to fulfill the underlying score FPE, which is an inherent self-consistency property of the ground truth score. We prove that satisfying the score 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.
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