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 and reverse process in the infinite-dimensional setting, 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：基于分数的扩散模型（SBDM）近期已成为图像生成领域的最先进方法。现有SBDM通常在有限维框架下构建，将图像视为有限尺寸的张量。本文在无限维框架下发展SBDM，即我们将训练数据建模为定义在矩形域上的函数。除了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，从而能够一致地在多个分辨率层级上进行离散化。我们期望通过此方法获得可跨不同分辨率层级泛化的扩散模型，并提升训练过程的效率。我们展示了如何克服当前SBDM方法在无限维框架下的两个缺陷：首先，修改前向过程，利用迹类算子概念确保隐变量分布在无限维框架下的良定性；其次，阐明使用算子网络（本文采用傅里叶神经算子（FNO））近似得分函数有利于多层级训练。在导出无限维框架下的前向与反向过程后，我们证明了其良定性，给出了适当的离散化方案，并探讨了隐变量分布的作用。我们在MNIST和材料结构两个数据集上获得了初步有前景的数值结果，特别展示了在此框架下多层级训练的可行性。