  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.
