Score-based diffusion models learn to reverse a stochastic differential equation that maps data to noise. However, for complex tasks, numerical error can compound and result in highly unnatural samples. Previous work mitigates this drift with thresholding, which projects to the natural data domain (such as pixel space for images) after each diffusion step, but this leads to a mismatch between the training and generative processes. To incorporate data constraints in a principled manner, we present Reflected Diffusion Models, which instead reverse a reflected stochastic differential equation evolving on the support of the data. Our approach learns the perturbed score function through a generalize score matching loss and extends key components of standard diffusion models including diffusion guidance, likelihood-based training, and ODE sampling. We also bridge the theoretical gap with thresholding: such schemes are just discretizations of reflected SDEs. On standard image benchmarks, our method is competitive with or surpasses the state of the art and, for classifier-free guidance, our approach enables fast exact sampling with ODEs and produces more faithful samples under high guidance weight.

翻译：基于分数的扩散模型学习逆推一种将数据映射为噪声的随机微分方程。然而，对于复杂任务，数值误差会累积并导致高度不自然的样本。先前的工作通过阈值化缓解这种偏移，即在每个扩散步骤后将结果投影到自然数据域（如图像的像素空间），但这会导致训练过程与生成过程不匹配。为以原则性方式纳入数据约束，我们提出反射扩散模型——该模型逆推一种在数据支撑集上演化的反射随机微分方程。通过广义分数匹配损失，我们的方法学习扰动分数函数，并扩展了标准扩散模型的关键组件，包括扩散引导、基于似然的训练以及常微分方程采样。我们还在理论上弥合了与阈值化的差距：此类方案本质上只是反射随机微分方程的离散化。在标准图像基准测试中，我们的方法可与当前最优方法竞争甚至超越之；对于无分类器引导，该方法能够通过常微分方程实现快速精确采样，并在高引导权重下生成更保真的样本。