Melt rates at the base of Antarctic ice shelves are needed to drive projections of the Antarctic ice sheet mass loss. Current basal melt parameterisations struggle to link open ocean properties to ice-shelf basal melt rates for the range of current sub-shelf cavity geometries around Antarctica. We present a novel parameterisation based on deep learning. With a simple feedforward neural network, or multilayer perceptron, acting on each grid cell separately, we emulate the behavior of circum-Antarctic cavity-resolving ocean simulations. We explore different neural network sizes and find that, in all cases containing at least one hidden layer, this kind of emulator produces reasonable basal melt rates for our training ensemble, closer to the reference simulation than traditional parameterisations. For testing, we use an independent ensemble of simulations that was produced with the same ocean model but with different model parameters, different cavity geometries and different forcing. In this challenging test, traditional and neural network parameterisations yield similar results on present conditions. In much warmer conditions than the training ensemble, both traditional parameterisations and neural networks struggle, but the neural networks tend to produce basal melt rates closer to the reference than a majority of traditional parameterisations. These neural networks are therefore suitable for century-scale Antarctic ice-sheet projections.