Denoising diffusion models have proven to be a flexible and effective paradigm for generative modelling. Their recent extension to infinite dimensional Euclidean spaces has allowed for the modelling of stochastic processes. However, many problems in the natural sciences incorporate symmetries and involve data living in non-Euclidean spaces. In this work, we extend the framework of diffusion models to incorporate a series of geometric priors in infinite-dimension modelling. We do so by a) constructing a noising process which admits, as limiting distribution, a geometric Gaussian process that transforms under the symmetry group of interest, and b) approximating the score with a neural network that is equivariant w.r.t. this group. We show that with these conditions, the generative functional model admits the same symmetry. We demonstrate scalability and capacity of the model, using a novel Langevin-based conditional sampler, to fit complex scalar and vector fields, with Euclidean and spherical codomain, on synthetic and real-world weather data.

翻译：去噪扩散模型已被证明是一种灵活且有效的生成建模范式。其近期向无限维欧几里得空间的扩展，使得对随机过程的建模成为可能。然而，自然科学中的许多问题都包含对称性，且涉及存在于非欧几里得空间中的数据。在本工作中，我们将扩散模型框架扩展至无限维建模中融入一系列几何先验。具体通过以下方式实现：a) 构建一个以在感兴趣对称群作用下变换的几何高斯过程为极限分布的加噪过程，以及b) 用相对于该群等变的神经网络近似分数函数。我们证明，在上述条件下，生成函数模型具有相同的对称性。通过采用一种新颖的基于朗之万动力学的条件采样器，我们在合成数据与真实天气数据上展示了该模型对具有欧几里得和球面余定义域的复杂标量场及向量场进行拟合的可扩展性与能力。