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According to the National Academies, a weekly forecast of velocity, vertical structure, and duration of the Loop Current (LC) and its eddies is critical for understanding the oceanography and ecosystem, and for mitigating outcomes of anthropogenic and natural disasters in the Gulf of Mexico (GoM). However, this forecast is a challenging problem since the LC behaviour is dominated by long-range spatial connections across multiple timescales. In this paper, we extend spatiotemporal predictive learning, showing its effectiveness beyond video prediction, to a 4D model, i.e., a novel Physics-informed Tensor-train ConvLSTM (PITT-ConvLSTM) for temporal sequences of 3D geospatial data forecasting. Specifically, we propose 1) a novel 4D higher-order recurrent neural network with empirical orthogonal function analysis to capture the hidden uncorrelated patterns of each hierarchy, 2) a convolutional tensor-train decomposition to capture higher-order space-time correlations, and 3) to incorporate prior physic knowledge that is provided from domain experts by informing the learning in latent space. The advantage of our proposed method is clear: constrained by physical laws, it simultaneously learns good representations for frame dependencies (both short-term and long-term high-level dependency) and inter-hierarchical relations within each time frame. Experiments on geospatial data collected from the GoM demonstrate that PITT-ConvLSTM outperforms the state-of-the-art methods in forecasting the volumetric velocity of the LC and its eddies for a period of over one week.