Ocean current, fluid mechanics, and many other physical systems with spatio-temporal dynamics are essential components of the universe. One key characteristic of such systems is that they can be represented by certain physics laws, such as ordinary/partial differential equations (ODEs/PDEs), irrespective of time or location. Physics-informed machine learning has recently emerged to learn physics from data for accurate prediction, but they often lack a mechanism to leverage localized spatial and temporal correlation or rely on hard-coded physics parameters. In this paper, we advocate a physics-coupled neural network model to learn parameters governing the physics of the system, and further couple the learned physics to assist the learning of recurring dynamics. Here a spatio-temporal physics-coupled neural network (ST-PCNN) model is proposed to achieve three goals: (1) learning the underlying physics parameters, (2) transition of local information between spatio-temporal regions, and (3) forecasting future values for the dynamical system. The physics-coupled learning ensures that the proposed model can be tremendously improved by using learned physics parameters, and can achieve useful long-range forecasting (e.g., more than two weeks). Experiments using simulated wave propagation and field-collected ocean current data validate that ST-PCNN outperforms typical deep learning models and existing physics-informed models.