Effective methods for inferring spatio-temporal models

This project focuses on improving computational efficiency for parameter estimation in complex spatiotemporal geostatistical models. Geostatistics aims to model natural phenomena evolving in space and time, and Gaussian processes are widely used to predict these phenomena in unobserved locations, while quantifying the uncertainty of the predictions. While classical models rely on simple parametric covariance functions, recent advances have introduced models based on stochastic partial differential equations (SPDEs), better suited to representing physical mechanisms such as advection or diffusion. These models become even more flexible when extended to non-stationary contexts, in which the properties of the process vary in space and time.

However, maximum likelihood parameter estimation quickly becomes expensive as the amount of data increases, a problem particularly pronounced for high-resolution spatiotemporal data. Although finite element discretization of SPDEs produces sparse matrices that accelerate likelihood calculations, optimization remains difficult for complex or non-stationary models. Conversely, these models are relatively easy to simulate, paving the way for an alternative strategy: simulation inference, or damped inference. This approach involves simulating a large number of spatiotemporal fields with known parameters and then training neural networks to either directly estimate these parameters or approximate the likelihood, thus enabling very fast inference.

This thesis will explore several complementary approaches to address these challenges. The first involves using automatic differentiation to optimize the likelihood of complex spatiotemporal models based on SPDEs, while developing flexible parameterizations for non-stationarity. A second approach builds upon preliminary work using graph neural networks (GNNs) for parameter inference in spatial fields; the goal will be to extend this approach to the spatiotemporal framework, which requires architectures capable of capturing neighborhood structures in space and time, as well as handling a much larger number of parameters.

Other approaches could also be considered, including hybrid models combining neural networks and Gaussian processes, variational inference, and Bayesian methods. The project will include a thorough comparison of the different methods based on their computational cost, accuracy, and scalability, as well as applications to real-world data from environmental sciences, such as solar radiation, wind speed, and air quality. The ultimate goal is to develop fast and reliable methods for estimating rich and physically realistic spatiotemporal models, useful for both methodological research and practical applications.