Score-based diffusion models (SBDM) have recently emerged as state-of-the-art approaches for image generation. Existing SBDMs are typically formulated in a finite-dimensional setting, where images are considered as tensors of a finite size. This papers develops SBDMs in the infinite-dimensional setting, that is, we model the training data as functions supported on a rectangular domain. Besides the quest for generating images at ever higher resolution our primary motivation is to create a well-posed infinite-dimensional learning problem so that we can discretize it consistently on multiple resolution levels. We thereby hope to obtain diffusion models that generalize across different resolution levels and improve the efficiency of the training process. We demonstrate how to overcome two shortcomings of current SBDM approaches in the infinite-dimensional setting. First, we modify the forward process to ensure that the latent distribution is well-defined in the infinite-dimensional setting using the notion of trace class operators. Second, we illustrate that approximating the score function with an operator network, in our case Fourier neural operators (FNOs), is beneficial for multilevel training. After deriving the forward process in the infinite-dimensional setting and reverse processes for finite approximations, we show their well-posedness, derive adequate discretizations, and investigate the role of the latent distributions. We provide first promising numerical results on two datasets, MNIST and material structures. In particular, we show that multilevel training is feasible within this framework.

翻译：分数扩散模型（SBDM）近期已成为图像生成领域最先进的方法。现有SBDM通常基于有限维设定，即将图像视为有限大小的张量。本文在无限维设定下发展SBDM，即将训练数据建模为定义在矩形区域上的函数。除追求生成更高分辨率图像外，我们的主要动机是构建一个适定的无限维学习问题，以便在多个分辨率层级上实现一致离散化。由此，我们期望获得能跨分辨率层级泛化的扩散模型，并提升训练过程效率。我们展示了如何克服当前SBDM方法在无限维设定中的两个缺陷：第一，通过引入迹类算子概念，修正前向过程以确保潜在分布在无限维设定中具有良定性；第二，阐明了使用算子网络（本文选用傅里叶神经算子FNO）近似分数函数对多尺度训练的有益性。在推导无限维前向过程及有限近似反向过程后，我们证明了其良定性，提出了适当的离散化方案，并研究了潜在分布的作用。我们在MNIST和材料结构两个数据集上给出了首批具有前景的数值结果，尤其验证了多尺度训练在此框架下的可行性。