The limb sounder radiometer on board the satellite Odin will be the first instrument measuring emission from space in the sub-millimeter region to map atmospheric species. Nonlinear inversions of Odin spectra by traditional iterative approaches will be computationally very intensive, so this paper proposes a faster inversion technique based on training neural networks. The technique is described first in general terms and then applied to invert simulated observations in two of the Odin bands, around 501.4 GHz and 544.6 GHz. To deal with the large dimension of the measured spectra, a data reduction based on the eigenvectors of the measured space is first applied. The reduced spectra are then input to a set of multilayer perceptrons that, after training with a set of simulated spectra, do the inversions. The same spectra are also inverted by optimal estimation, and the performance from both techniques is compared. The neural network technique retrieves species profiles with errors and vertical resolutions comparable to optimal estimation, it can be made very robust against the uncertainties of the a priori information by including different learning terms during the training, and it is faster than optimal estimation if nonlinear inversions are required. Although final conclusions on processing time have to wait until Odin is operational, the simulations show that the neural network technique has the potential to make the Odin nonlinear inversions at least 1 order of magnitude faster than they are made by optimal estimation.