We introduce a formal active learning methodology for guiding the placement of Lagrangian observers to infer time-dependent vector fields -- a key task in oceanography, marine science, and ocean engineering -- using a physics-informed spatio-temporal Gaussian process surrogate model. The majority of existing placement campaigns either follow standard `space-filling' designs or relatively ad-hoc expert opinions. A key challenge to applying principled active learning in this setting is that Lagrangian observers are continuously advected through the vector field, so they make measurements at different locations and times. It is, therefore, important to consider the likely future trajectories of placed observers to account for the utility of candidate placement locations. To this end, we present BALLAST: Bayesian Active Learning with Look-ahead Amendment for Sea-drifter Trajectories. We observe noticeable benefits of BALLAST-aided sequential observer placement strategies on both synthetic and high-fidelity ocean current models. In addition, we developed a novel GP inference method -- the Vanilla SPDE Exchange (VaSE) -- to boost the GP posterior sampling efficiency, which is also of independent interest.

翻译：本文提出一种形式化的主动学习方法，用于指导拉格朗日观测器的布放，以推断时变矢量场——这是海洋学、海洋科学与海洋工程中的关键任务。该方法采用物理信息时空高斯过程代理模型。现有的大多数布放方案要么遵循标准的“空间填充”设计，要么依赖相对临时的专家意见。在此背景下应用原则性主动学习的主要挑战在于：拉格朗日观测器在矢量场中持续平流运动，因此会在不同位置和时间进行测量。因此，必须考虑已布放观测器的可能未来轨迹，以评估候选布放位置的效用。为此，我们提出BALLAST：面向海漂轨迹的贝叶斯主动学习前瞻修正方法。我们在合成和高保真洋流模型上均观察到BALLAST辅助的序贯观测器布放策略带来的显著优势。此外，我们开发了一种新颖的高斯过程推断方法——Vanilla SPDE Exchange（VaSE）——以提升高斯过程后验采样效率，该方法亦具有独立的研究价值。