First inversions of observed limb sounding data by a very fast neural network technique are presented. The technique is based on training a set of multilayer perceptrons. The training set was built by random generation of atmospheric states followed by forward model calculations to derive the corresponding radiances. A novel spectral reduction technique was used to reduce the dimensionality of the spectral space, allowing the setting up of multilayer perceptrons with a practical topology. For training, an algorithm that relies on Bayesian techniques was applied in order to favor generalization. The technique was tested by inverting measured radiances from the Odin-SMR submillimeter radiometer and judged by comparing with similar inversions from an optimal estimation code. The neural network technique resulted in much faster inversions than the optimal estimation code, but biases between the optimal estimation and neural network retrievals were found. The problem was tracked down to the presence of spectral artifacts in the measured radiances. The optimal estimation inversions were also affected but seemed less sensitive. Nevertheless, the results were promising and opened the possibility of implementing a very fast processing algorithm for the Odin-SMR data.