  The reconstruction of gap-free signals from observation data is a critical
challenge for numerous application domains, such as geoscience and space-based
earth observation, when the available sensors or the data collection processes
lead to irregularly-sampled and noisy observations. Optimal interpolation (OI),
also referred to as kriging, provides a theoretical framework to solve
interpolation problems for Gaussian processes (GP). The associated
computational complexity being rapidly intractable for n-dimensional tensors
and increasing numbers of observations, a rich literature has emerged to
address this issue using ensemble methods, sparse schemes or iterative
approaches. Here, we introduce a neural OI scheme. It exploits a variational
formulation with convolutional auto-encoders and a trainable iterative
gradient-based solver. Theoretically equivalent to the OI formulation, the
trainable solver asymptotically converges to the OI solution when dealing with
both stationary and non-stationary linear spatio-temporal GPs. Through a
bi-level optimization formulation, we relate the learning step and the
selection of the training loss to the theoretical properties of the OI, which
is an unbiased estimator with minimal error variance. Numerical experiments for
2D+t synthetic GP datasets demonstrate the relevance of the proposed scheme to
learn computationally-efficient and scalable OI models and solvers from data.
As illustrated for a real-world interpolation problems for satellite-derived
geophysical dynamics, the proposed framework also extends to non-linear and
multimodal interpolation problems and significantly outperforms
state-of-the-art interpolation methods, when dealing with very high missing
data rates.
