  The spatio-temporal interpolation of large geophysical datasets has
historically been adressed by Optimal Interpolation (OI) and more sophisticated
model-based or data-driven DA techniques. In the last ten years, the link
established between Stochastic Partial Differential Equations (SPDE) and
Gaussian Markov Random Fields (GMRF) opened a new way of handling both large
datasets and physically-induced covariance matrix in Optimal Interpolation.
Recent advances in the deep learning community also enables to adress this
problem as neural architecture embedding data assimilation variational
framework. The reconstruction task is seen as a joint learning problem of the
prior involved in the variational inner cost and the gradient-based
minimization of the latter: both prior models and solvers are stated as neural
networks with automatic differentiation which can be trained by minimizing a
loss function, typically stated as the mean squared error between some ground
truth and the reconstruction. In this work, we draw from the SPDE-based
Gaussian Processes to estimate complex prior models able to handle
non-stationary covariances in both space and time and provide a stochastic
framework for interpretability and uncertainty quantification. Our neural
variational scheme is modified to embed an augmented state formulation with
both state and SPDE parametrization to estimate. Instead of a neural prior, we
use a stochastic PDE as surrogate model along the data assimilation window. The
training involves a loss function for both reconstruction task and SPDE prior
model, where the likelihood of the SPDE parameters given the true states is
involved in the training. Because the prior is stochastic, we can easily draw
samples in the prior distribution before conditioning to provide a flexible way
to estimate the posterior distribution based on thousands of members.
