  Gaussian processes (GPs) provide a principled and direct approach for
inference and learning on graphs. However, the lack of justified graph kernels
for spatio-temporal modelling has held back their use in graph problems. We
leverage an explicit link between stochastic partial differential equations
(SPDEs) and GPs on graphs, introduce a framework for deriving graph kernels via
SPDEs, and derive non-separable spatio-temporal graph kernels that capture
interaction across space and time. We formulate the graph kernels for the
stochastic heat equation and wave equation. We show that by providing novel
tools for spatio-temporal GP modelling on graphs, we outperform pre-existing
graph kernels in real-world applications that feature diffusion, oscillation,
and other complicated interactions.
