Estimation of spatio-temporal model by deep learning
In geostatistics, Gaussian processes are commonly used to model spatial and spatio-temporal data because they allow for straightforward prediction of the variable of interest at unmeasured sites while quantifying the uncertainty of the prediction. In this context, the data are considered to be a realization of a Gaussian random field, whose covariance function needs to be estimated from the data. A classical approach for inference is to select a parameterized family of covariance functions and then choose the parameters that maximize the likelihood associated with the data. In practice, this approach often becomes a bottleneck, as even evaluating the likelihood function can quickly become computationally expensive when dealing with large amounts of data, especially in spatio-temporal settings. Therefore, it is desirable to have methods that allow for deducing the parameters of a covariance model without relying on the likelihood function.
Recently, several methods using neural networks (notably CNNs and GNNs) have been proposed to address this issue. The main approach aims to train a neural network capable of identifying the parameters of a covariance function based on a realization of a Gaussian random field with that covariance (Gerber and Nychka 2021; Lenzi et al. 2023; Sainsbury-Dale et al. 2023). It can be considered a Simulation-Based Inference (SBI) approach, also called amortized inference.
First step : Vincent Fourmigué’s internship
A first step in this direction has been made by the M1 student Antoine Regardin during a 3-month internship in 2024, funded by the Geolearning chair.
This internship aimed to develop and validate the GNN architecture proposed in Sainsbury-Dale et al. (2023), adapted to the spatial context. The architecture was coded in PyTorch with the library PyG(PyTorch_Geometric).
Current work : Alexandre Loret’s PhD
Alexandre Loret has begun his PhD in october 2025 and he will address the generalization of the SBI approach to a setting adapted to spatio-temporal data. The main challenges of this generalization include designing a GNN architecture that accounts for spatio-temporal neighborhood structures and handling the estimation of a significantly larger number of parameters. He will also apply the approach to models derived from stochastic partial differential equations (Clarotto et al. 2024).
More generally, he will tackle the computational challenges of the inference of complex spatio-temporal models, exploring different approaches. Several possibilities exist to address this problem, among which hybrid models that integrate neural networks within Gaussian processes, such as Deep Gaussian Markov Fields, as well as variational inference methods or Neural Operators.