Nonlocal Bayesian Modeling of Continuous Spatio-Temporal Dynamics

Real-world spatio-temporal forecasting must handle irregular time points, spatially sparse observations, and the need for uncertainty quantification. This setting is often further compounded by nonlocal interactions (long-range spatial coupling). Modeling continuous-space, continuous-time nonlocal dynamics naturally leads to infinite-dimensional integro-differential equations (IDEs), making principled {Bayesian} inference intractable. We propose the NonLocal {Bayesian} Spatio-Temporal model (NLBST), a hierarchical {Bayesian} framework for continuous spatio-temporal fields that learns explicit nonlocal coupling while retaining tractable inference. NLBST represents the latent field via a coordinate-based spatial basis expansion and models the coefficient process with a continuous-time {ODE} whose learnable linear operator corresponds to a Galerkin reduction of a nonlocal IDE; a Neural {ODE} residual captures additional nonlinear dynamics. A linear-{Gaussian} observation model enables {Kalman}-style sequential updates under missing and irregular observations, while the spatial basis representation enables inductive prediction at unmeasured locations without retraining. Global parameters are learned via variational inference, and uncertainty is handled through a {Bayesian} hierarchy. Experiments on synthetic and real-world datasets demonstrate strong forecasting and spatial generalization with well-calibrated uncertainty, yielding substantial gains over baselines in strongly nonlocal and partially observed regimes.