Terrestrial evaporation (E) is a key climatic variable that depends on a plethora of environmental factors. The constraints that modulate the evaporation from plant leaves (or transpiration, Et) are particularly complex, yet often assumed to interact linearly in global models due to our limited knowledge based on local experimental studies. Here, we combine in situ and satellite observations with deep learning to model transpiration stress (St), i.e. the reduction of Et from its theoretical maximum. Then, we embed the new St formulation within a process-based model of E to yield a global hybrid E model. In this hybrid, the St formulation is bidirectionally coupled to the the host model at daily timescales. Comparisons against in situ data and satellite-based proxies demonstrate an enhanced ability to estimate St and E globally. Therefore, the proposed approach provides a framework to improve the estimation of E in Earth System Models and our understanding of this crucial climatic variable.