Estimation of spatio-temporal model by deep learning
As part of a collaboration with Thomas Romary and Mike Pereira (Mines Paris), we proposed an Master1-level internship (three months) focused on parameter estimation for spatio-temporal models using graph neural networks. The internship was funded by the [_Geolearning_] chair and took place from June to August 2023 with Vincent Fourmigué, a student at AgroParisTech. Although the student was employed by Mines Paris, he spent nearly the entire internship at the UMR MIA Paris-Saclay facilities on the Agro Paris Saclay Campus.
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 et al., 2022, Lenzi et al., 2023, Wikle et al., 2022, Sainsbury-Dale et al., 2023).
This internship aimed to develop and validate an architecture adapted to the spatio-temporal context. One challenge of this generalization is the estimation of a larger number of parameters, which has not been implemented so far. The first step involved coding (using the PyGeometric library in Python) the strategy highlighted in the literature, initially on simulated datasets. In the second phase, the goal was to adapt this approach to the spatio-temporal case, particularly for models derived from stochastic partial differential equations (Clarotto et al., 2024).