We discuss improving forecasts of winds in the lower stratosphere using machine learning to post-process the output of the European Centre for Medium-Range Weather Forecasts (ECMWF) Integrated Forecast System. We post-process global three-dimensional predictions, and demonstrate distilling the analog ensemble (AnEn) method into a deep neural network which reduces post-processing latency to near zero maintaining increased forecast skill. This approach reduces the error with respect to ECMWF high-resolution deterministic prediction between 2-15% for wind speed and 15-25% for direction, and is on par with ECMWF ensemble (ENS) forecast skill to hour 60. Verifying with Loon data from stratospheric balloons, AnEn has 20% lower error than ENS for wind speed and 15% for wind direction, despite significantly lower real-time computational cost to ENS. Similar performance patterns are reported for probabilistic predictions, with larger improvements of AnEn with respect to ENS. We also demonstrate that AnEn generates a calibrated probabilistic forecast.