Climate risk assessments typically focus on large rivers, but water availability and excess critically depend on catchment area. The enigmatic yet paramount scaling relationships have for decades remained poorly understood and thus under-represented in Earth System models. With state-of-the-art physics-informed artificial intelligence, we learned scaling relationships from >3000 basins, highlighting overlooked climate risks. Catchment-area scaling has outsized impacts on mean specific water supply (arid-basin median ~25%), groundwater ratio (~30%), and runoff climate sensitivities to temperature (>200%) and precipitation (~10%), with distinct regional patterns. In arid regions, as catchment area increases, mean streamflow supply declines by 25% (median) at ~50km length scale (increasing with aridity) when groundwater return flow balances stream water loss. Floods in headwater catchments have larger sensitivity to precipitation intensification, while baseflows downstream are more susceptible to temperature changes. Physics-informed learning demonstrates a systematic solution to spatial scaling and elucidates the fine-scale distribution of climate risks across communities.