Physics-augmented Multi-task Gaussian Process for Modeling Spatiotemporal Dynamics

Abstract

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. Note to Practitioners—This article proposes a P-G-MGP framework designed for predictive modeling of spatiotemporal systems over complex geometric domains. A central feature of the proposed approach is its ability to capture interrelated physical variables while respecting underlying dynamical laws. This framework can provide more accurate and physically consistent predictions even when available data are limited or irregularly distributed. Although our case study focuses on 3D cardiac electrodynamics, the framework is broadly applicable to other engineering and scientific domains where accurate spatiotemporal prediction is critical, such as environmental monitoring, structural integrity assessment, fluid dynamics, and advanced manufacturing. By fusing geometry priors, multi-task relationships, and physics-based regularization, our framework achieves higher predictive fidelity and greater robustness in the presence of sparse or noisy measurements. These capabilities enable reliable decision support in real-world systems where the interplay of geometry and physics is indispensable.