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 currents and water supply networks, comparing them to alternative models.

翻译：本文提出了一种严谨的高斯过程（GPs）方法，用于建模定义在单纯2-复形（一种类似于图的结构，其中边可能形成三角形面）边集上的函数。该方法旨在学习网络上的流型数据，其中边流可通过离散散度和旋度进行表征。基于霍奇分解，我们首先开发了适用于多种场景的无散度和无旋度边高斯过程类。随后，通过组合这两类过程，我们构建了具有足够表达能力以表示任意边函数的**霍奇组合边高斯过程**。这些高斯过程能够直接且独立地学习边函数的不同霍奇分量，从而在超参数优化过程中捕获其相关性。为突出其实用潜力，我们将该方法应用于汇率、洋流及供水网络中的流数据推断，并与替代模型进行了比较。