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This seems like a natural fit, though I've not heard of any, yet.

I would love to know if any MET office, government, military or academic institution has taken all (or sizeable portion of) recorded global weather data for, say, the last 50 years (or since we, as a race, have been using weather satellites) and used it in an AI system to predict future weather.

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People have used machine learning models on aspects of weather forecasting, as here: Convolutional LSTM Network: A Machine Learning Approach for Precipitation Nowcasting or here Predicting Solar Generation from Weather Forecasts using Machine Learning. I've been loosely associated with an effort to use ML techniques to predict utility demand from weather data. But note that these are looking to predict the implications of the weather forecast, not the forecast itself. Lots of effort goes into refining the physics-based models that underlie our everyday weather forecasts, and AFAIK machine learning hasn't resulted in any "secret sauce" that gives better results than these models provide.

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