The tokamak is currently the principal experimental system for research into the magnetic confinement approach to controlled fusion. Hydrogen gas is raised to very high temperatures inside a toroidal vacuum vessel, and the resulting plasma is confined by a complex system of magnetic fields. Measurements of the electron density inside a tokamak can be made using laser interferometry, which gives line-integral information along chords through the plasma. Extraction of spatially local information from this line integral data represents an ill-posed inverse problem. In this paper we describe a novel approach to the solution of this problem, based on feedforward networks, and we show that it leads to improved accuracy of reconstruction compared with conventional techniques. A software implementation of the trained network has been installed at JET and will be used on a routine basis for profile reconstruction.