To unlock access to stronger winds, the offshore wind industry is advancing towards significantly larger and taller wind turbines. This massive upscaling motivates a departure from wind forecasting methods that traditionally focused on a single representative height. To fill this gap, we propose DeepMIDE--a statistical deep learning method which jointly models the offshore wind speeds across space, time, and height. DeepMIDE is formulated as a multi-output integro-difference equation model with a multivariate nonstationary kernel characterized by a set of advection vectors that encode the physics of wind field formation and propagation. Embedded within DeepMIDE, an advanced deep learning architecture learns these advection vectors from high-dimensional streams of exogenous weather information, which, along with other parameters, are plugged back into the statistical model for probabilistic multi-height space-time forecasting. Tested on real-world data from offshore wind energy areas in the Northeastern United States, the wind speed and power forecasts from DeepMIDE are shown to outperform those from prevalent time series, spatio-temporal, and deep learning methods.