  Historically, the interpolation of large geophysical datasets has been
tackled using methods like Optimal Interpolation (OI) or model-based data
assimilation schemes. However, the recent connection between Stochastic Partial
Differential Equations (SPDE) and Gaussian Markov Random Fields (GMRF)
introduced a novel approach to handle large datasets making use of sparse
precision matrices in OI. Recent advancements in deep learning also addressed
this issue by incorporating data assimilation into neural architectures: it
treats the reconstruction task as a joint learning problem involving both prior
model and solver as neural networks. Though, it requires further developments
to quantify the associated uncertainties. In our work, we leverage SPDEbased
Gaussian Processes to estimate complex prior models capable of handling
nonstationary covariances in space and time. We develop a specific architecture
able to learn both state and SPDE parameters as a neural SPDE solver, while
providing the precisionbased analytical form of the SPDE sampling. The latter
is used as a surrogate model along the data assimilation window. Because the
prior is stochastic, we can easily draw samples from it and condition the
members by our neural solver, allowing flexible estimation of the posterior
distribution based on large ensemble. We demonstrate this framework on
realistic Sea Surface Height datasets. Our solution improves the OI baseline,
aligns with neural prior while enabling uncertainty quantification and online
parameter estimation.
