Dynamic Gaussian Processes and the Vanilla-SPDE Exchange

Gaussian process (GP) inference is traditionally limited by cubic computational complexity, which becomes particularly problematic in spatio-temporal settings. While standard static GP formulations struggle with large datasets, dynamic SPDE-based formulations offer linear scaling in time but often suffer from high spatial costs when observation and prediction locations are disjoint. This paper asks whether a hybrid approach can combine the strengths of both formulations to achieve more efficient inference.

The authors propose the Vanilla-SPDE Exchange (VaSE), a hybrid inference technique. VaSE performs the initial regression step using the standard static GP formulation, which is efficient for handling observations. Once the posterior distribution is obtained at the forecast boundary, the method converts this posterior into an SPDE state representation. This state is then used to propagate future predictions and generate posterior samples sequentially using the dynamic SPDE formulation. By decomposing the inference process, VaSE avoids the high costs associated with forcing a single formulation to handle both regression and long-term forecasting in disjoint spatial settings.

VaSE provides a flexible framework that outperforms both pure static and pure dynamic methods in scenarios where observation and prediction locations are spatially disjoint. The authors demonstrate that while static GPs are efficient for regression and SPDE-based GPs are efficient for sequential sampling, VaSE captures the best of both worlds. Numerical experiments confirm that VaSE maintains lower computational walltime and memory usage compared to existing methods as the number of observations increases, particularly when predictions are required on dense spatio-temporal grids.

This research provides a practical solution for researchers working with spatio-temporal data who need to perform exact inference without the prohibitive costs of standard GP methods. By allowing practitioners to switch between static and dynamic representations, VaSE enables more scalable posterior sampling and forecasting, making it a valuable tool for complex spatio-temporal modeling tasks.