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Current climate models require representations of processes that occur at scales smaller than model grid size, which have been the main source of model projection uncertainty. Recent machine learning (ML) algorithms hold promise to improve such process representations, but tend to extrapolate poorly to climate regimes they were not trained on. To get the best of the physical and statistical worlds, we propose a new framework - termed \"climate-invariant\" ML - incorporating knowledge of climate processes into ML algorithms, and show that it can maintain high offline accuracy across a wide range of climate conditions and configurations in three distinct atmospheric models. 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