We propose principled Gaussian processes (GPs) for modeling functions defined over the edge set of a simplicial 2-complex, a structure similar to a graph in which edges may form triangular faces. This approach is intended for learning flow-type data on networks where edge flows can be characterized by the discrete divergence and curl. Drawing upon the Hodge decomposition, we first develop classes of divergence-free and curl-free edge GPs, suitable for various applications. We then combine them to create \emph{Hodge-compositional edge GPs} that are expressive enough to represent any edge function. These GPs facilitate direct and independent learning for the different Hodge components of edge functions, enabling us to capture their relevance during hyperparameter optimization. To highlight their practical potential, we apply them for flow data inference in currency exchange, ocean flows and water supply networks, comparing them to alternative models.

翻译：我们提出了用于定义在单纯2-复形边集（一种类似于图且边可构成三角面的结构）上函数建模的严谨高斯过程。该方法旨在对网络上的流型数据进行学习，其中边流可通过离散散度和旋度进行表征。基于Hodge分解，我们首先开发了适用于多种应用场景的无散度和无旋度边高斯过程类。随后将这些过程组合成表达能力足以表示任意边函数的**Hodge分解式边高斯过程**。该过程可对边函数的不同Hodge分量进行直接且独立的学习，从而在超参数优化过程中捕获各分量的相关性。为凸显其实用潜力，我们将该方法应用于货币汇率、海洋环流及供水网络中的流数据推断，并与替代模型进行了比较。