# A diffusion-based spatio-temporal extension of Gaussian Mat'ern fields

arXiv: https://arxiv.org/abs/2006.04917 · HTML: https://mlsanity.com/paper/2006.04917/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Jun 2020.

## Built on by (most cited first, 2 of 2)

- [Non-separable Covariance Kernels for Spatiotemporal Gaussian Processes based on a Hybrid Spectral Method and the Harmonic Oscillator](https://mlsanity.com/paper/2302.09580/) (2023)
- [Flexible and Efficient Probabilistic PDE Solvers through Gaussian Markov Random Fields](https://mlsanity.com/paper/2503.08343/) (2025)

## Abstract (arXiv)

Gaussian random fields with Mat'ern covariance functions are popular models in spatial statistics and machine learning. In this work, we develop a spatio-temporal extension of the Gaussian Mat'ern fields formulated as solutions to a stochastic partial differential equation. The spatially stationary subset of the models have marginal spatial Mat'ern covariances, and the model also extends to Whittle-Mat'ern fields on curved manifolds, and to more general non-stationary fields. In addition to the parameters of the spatial dependence (variance, smoothness, and practical correlation range) it additionally has parameters controlling the practical correlation range in time, the smoothness in time, and the type of non-separability of the spatio-temporal covariance. Through the separability parameter, the model also allows for separable covariance functions. We provide a sparse representation based on a finite element approximation, that is well suited for statistical inference and which is implemented in the R-INLA software. The flexibility of the model is illustrated in an application to spatio-temporal modeling of global temperature data.