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.

翻译：现实世界的时空预测必须处理不规则时间点、空间稀疏观测以及对不确定性量化的需求。这一场景通常因非局部相互作用（长程空间耦合）而进一步复杂化。对连续空间、连续时间的非局部动力学进行建模，自然会引出无限维积分微分方程，这使得基于原理的贝叶斯推断变得棘手。我们提出非局部贝叶斯时空模型，这是一种用于连续时空场的分层贝叶斯框架，能够在保持可处理推断的同时学习显式的非局部耦合。该模型通过基于坐标的空间基扩展表示潜在场，并使用连续时间常微分方程对系数过程建模，该常微分方程的可学习线性算子对应于非局部积分微分方程的伽辽金约化；神经常微分方程残差则捕获额外的非线性动力学。线性高斯观测模型支持在缺失和不规则观测情况下进行类卡尔曼的序贯更新，而空间基表示则无需重新训练即可对未测量位置进行归纳预测。全局参数通过变分推断学习，不确定性则通过贝叶斯层级结构处理。在合成数据集和真实数据集上的实验表明，该模型具有强大的预测能力和空间泛化能力，且不确定性校准良好，在强非局部和部分观测场景下显著优于基线方法。