Gaussian Graph(ical) Neural Networks for repeated observations over a fixed graph with persistent edge relationships

Many networked systems are observed repeatedly over a fixed physical or relational structure. In applications such as environmental monitoring, individual observations may be sparse, noisy, or incomplete, making the current state difficult to estimate from measurements alone. However, repeated observations provide an opportunity to learn dependencies between nodes that persist across measurements and can constrain future inference. Existing graph neural networks do not naturally encode such dependencies, while classical Gaussian graphical models encode them explicitly but offer limited representational flexibility. We introduce Gaussian Graphical Neural Networks, which combine expressive neural representations with Gaussian graphical inference by learning a sparse precision matrix over the fixed graph. We develop three precision-matrix parameterizations that guarantee convergent inference, characterize the spectral behavior of the resulting operator and show that it imposes no inherent smoothing bias, and derive an implicit differentiation procedure that enables end-to-end training without unrolling inference. Experiments on spatial reconstruction and hydrological forecasting demonstrate that learning persistent graphical dependencies and performing structured inference over latent representations can improve estimation in repeated-observation settings.