In this article , Yongquan Qu, Mohamed Aziz Bhouri, and Pierre Gentine, tackle the joint inference and uncertainty quantification of poorly known physical parameters and machine learning parametrizations, for sub-grid scale dynamics. Achieved through a Bayesian framework enabled by differentiable programming, this method not only offers accurate parameter estimates, but also enables skillful forecasting, each accompanied by quantified uncertainty. This work is accepted at the ICLR 2024 Workshop on AI4Differential Equations In Science.