A fundamental challenge in the design of LEDs is to maximise electro-luminescence efficiency at high current densities. We simulate GaN-based LED structures that delay the onset of efficiency droop by spreading carrier concentrations evenly across the active region. Statistical analysis and machine learning effectively guide the selection of the next LED structure to be examined based upon its expected efficiency as well as model uncertainty. This active learning strategy rapidly constructs a model that predicts Poisson-Schrödinger simulations of devices, and that simultaneously produces structures with higher simulated efficiencies.