Non-separable Spatio-temporal Graph Kernels via SPDEs

TL;DR

This work tackles the absence of principled spatio-temporal graph kernels for Gaussian processes by deriving kernels from stochastic partial differential equations on graphs. It introduces a general SPDE-to-graph framework and develops two non-separable kernels, SHEK and SWEK, from the stochastic heat and wave equations, respectively. Through experiments on heat diffusion and epidemic data (COVID-19 and chickenpox), the non-separable kernels demonstrate competitive or superior performance and improved robustness over time, illustrating the value of physically informed priors for graph-based temporal problems. The approach broadens the toolbox for probabilistic graph modelling, enabling more accurate, uncertainty-aware predictions for diffusion and oscillatory dynamics on networks, with potential applications in epidemiology and infrastructure systems.

Abstract

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.

Figures (11)

• Figure 1: We start with an unknown process on a particular domain with an SPDE model, discretize it, and derive the GP covariance function from this SPDE. Finally, we use GP inference in the spatial or spatio-temporal model from discrete graph-structured observations.
• Figure 2: Temporal visualizations of SHEK (heat, left) and SWEK (wave, right) on a linear three-node graph. The first row shows the temporal part of the covariance matrix (summed over the graph vertices at each timepoint). The following three rows show mean (black) and samples (colored lines) as a function of time at each of the nodes, conditioned on $y(t=0) = (0, 0, 10)$, for different values of the hyperparameter $c$.
• Figure 3: Visualization of extrapolation evaluation with extended extrapolation periods of four and six weeks on COVID-19 dataset.
• Figure S1: Temporal visualizations of SHEK (heat, left) and SWEK (wave, right) on a linear three-node graph. The first row shows the temporal part of the covariance matrix (summed over the graph vertices at each timepoint). The following three rows show mean (black) and samples (colored lines) as a function of time at each of the nodes, conditioned on $y(t=0) = (0, 0, 10)$, for different values of the hyperparameter $c$.
• Figure S2: As in \ref{['figure:appendix_graph_visualizations']}, here we illustrate mean and marginal variance.

Theorems & Definitions (22)

• Definition 3.1
• Definition 4.1
• Proposition 1
• Proposition 2
• Proposition 3
• Theorem A.1: Itô isometry, oksendal2013stochastic
• Corollary A.1.1
• Lemma A.2: Integration of the matrix exponential
• Proposition B.1: Laplace's equation in kernel form
• proof