The monsoonal precipitation is dominated by intraseasonal variabilities, whose prediction lead time is no longer than 5 days but still remains a grand challenge. Here we show that a combination of dynamics of air-sea interactions and a machine learning (ML) algorithm demonstrates a skillful prediction of precipitation over the monsoon region with a lead time of over 15 days. This prediction capability is significantly longer than the current ability and close to its theoretical predictability limit. The dynamics maintain the physical constraints on the intraseasonal monsoon rainfall, while the data-driven ML algorithm suppresses the unwanted high-frequency noises without a physical constraint. The physics- and computation-based forecast system hybridizes physical fundamentals and the versatility of a data-driven algorithm. The identification of the coupled air-sea processes and the verification of their contributions to the significant increase in intraseasonal prediction advances our understanding of the coupled monsoon system and represents the potential for improved simulations, as well as useful and usable predictions of monsoon precipitation. In addition, the high prediction skill demonstrates a prospective application of the booming ML techniques in weather forecasts and climate projections.