Prediction of spatio-temporal data on meshed surfaces using advection-diffusion SPDEs
Statisticians analyzing environmental data have recently shown great interest in introducing models inspired by the physics of underlying phenomena to improve prediction methods in a spatio-temporal context. The statistical method based on Stochastic Partial Differential Equations (SPDE) is an innovative approach to simulate, estimate, and predict spatial and spatio-temporal fields. A comprehensive and detailed formalism of the SPDE approach in the spatial context was introduced by Lindgren et al. (2011). Significant mathematical and algorithmic advances have been made over the past decade, enabling efficient handling of very large datasets Pereira et al. (2019). Moreover, Vergara et al. (2022) extended the approach to the spatio-temporal framework by incorporating physical processes related to the phenomena under study.
The SPDE approach relies on approximating a continuously indexed Gaussian random field (GRF) as a discretely indexed random process, specifically a Gaussian Markov random field (GMRF). Transitioning from a GRF to a GMRF replaces the dense covariance function and matrix with a neighborhood structure and a sparse precision matrix, respectively. The use of GMRFs with sparse precision matrices leads to computationally efficient numerical methods.
To better describe phenomena characterized by diffusive behavior that also exhibits transport in a preferred direction, I proposed in my thesis, completed in 2023, the class of advection-diffusion SPDEs:
\[ \left[\frac{\partial}{\partial t} + \frac{1}{c}(\kappa^2 - \nabla \cdot \mathbf{H}\nabla)^{\alpha} + \frac{1}{c}\mathbf{\gamma} \cdot \nabla \right] u = \frac{\tau}{\sqrt{c}} \mathcal{Z}, \]
where \(\nabla \cdot \mathbf{H}\nabla\) is a diffusion operator with possible anisotropy \(\mathbf{H}\), \(\mathbf{\gamma} \cdot \nabla\) is an advection operator, \(\alpha\geq0\) governs the regularity of \(u\), \(\kappa^2>0\) is a damping term related to the spatial range of \(u\), and \(c\) is a time scale parameter. Finally, \(\tau\geq 0\) is a standard deviation. The stochastic nature arises from the stochastic forcing term \(\mathcal{Z}\), which can be a spatio-temporal white noise \[\mathcal{Z} = \mathcal{W}_S \otimes \mathcal{W}_T\] or a colored noise.
The different terms of the SPDE (advection, diffusion) directly influence the spatio-temporal dependencies of the process by controlling its variability in space and time.
Through finite element and finite difference discretization of the SPDE, this approach leads to a sparse structure in the precision matrix of the spatio-temporal field, allowing the use of fast algorithms for estimating SPDE parameters and for spatio-temporal prediction using kriging.
Compared to spatio-temporal models built using covariance kernels, this model gains not only in computational efficiency but also in interpretability, as the model parameters can be linked to the physical coefficients of the SPDE.
The objective of this new work, in collaboration with Mike Pereira and Nicolas Desassis, is to propose a statistical model for spatio-temporal data on meshed surfaces based on the SPDE modeling approach. Specifically, we focus on a class of advection-diffusion SPDEs defined on smooth compact orientable closed Riemannian manifolds of dimension 2, and their discretization via a Galerkin approach. We demonstrate how this method enables the development of scalable algorithms for the simulation and prediction of Gaussian random fields that are solutions to the discretized SPDE. Additionally, we present recent developments in the inference of such models. The method is applied to a simulated spatio-temporal dataset exhibiting advective and diffusive behavior on the sphere, as well as to a real case study on aerosol optical depth in the atmosphere across the globe’s surface.
More details can be found in Clarotto et al. (2024) and Pereira, Clarotto, and Desassis (2025).