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In this paper, we show that a revised convolutional recurrent neural network (CRNN) can decrease, by orders of magnitude, the time needed for the phase-resolved prediction of waves in a spatiotemporal domain of a nonlinear dispersive wave field. The problem of predicting such waves suffers from two major challenges that have so far hindered analytical or direct computational solutions in real time or faster: (i) the reconstruction problem, that is, how one can calculate from measurable wave amplitude data the state of the wave field (wave components, nonlinear couplings, etc.), and (ii) if such a reconstruction is in hand, how to integrate equations fast enough to be able to predict an upcoming rouge wave in a timely manner. Here, we demonstrate that these two challenges can be overcome at once through advanced machine learning techniques based on spatiotemporal patches of the time history of wave height data in the domain. Specifically, as a benchmark here we consider equations that govern the evolution of weakly nonlinear surface gravity waves such as those propagating on the surface of the oceans. For the case of oceanic surface waves considered here, we demonstrate that the proposed methodology, while maintaining a high accuracy, can make phase-resolved predictions more than two orders of magnitude faster than numerically integrating governing equations.