Physics-Augmented Multi-Task Gaussian Process for Modeling Spatiotemporal Dynamics

Recent advances in sensing and imaging technologies have enabled the acquisition of high-dimensional spatiotemporal data from complex geometric domains. However, predictive modeling of such systems remains challenging due to irregular spatial manifolds, coupled multi-output dynamics, limited observations, and the need for reliable uncertainty quantification. This paper presents a physics-augmented, geometry-aware, multi-task Gaussian Process (P-G-MGP) framework for spatiotemporal modeling. We develop a geometry-aware multi-task GP (G-MGP) to capture spatiotemporal structures and inter-task dependencies. To enhance model fidelity and robustness, we incorporate governing physical laws through a physics-based regularization scheme, thereby constraining predictions to be consistent with governing principles. Furthermore, our framework provides closed-form estimates of posterior variance, enabling calibrated uncertainty quantification for downstream decision-making. We validate P-G-MGP on 3D cardiac electrophysiological modeling, demonstrating superior predictive performance over existing methods by effectively incorporating geometric priors, multi-task interactions, and domain-specific physical constraints.