Inverse scattering methods for reconstructing sound-wave-speed structure in the dimensions have been shown to be equivalent to inverting line integrals when the scattered field is of sufficiently high frequency and the scattering is sufficiently weak. Seismic traveltime tomography uses first arrival traveltime data to invert for wave-speed structure. Of course, the traveltime is itself a line integral along a refracting ray path through the medium being probed. The similarity between these two inversion problems is discussed. One type of neural network-the Hopfield net-may be used to improve the traveltime inversion. We find that, by taking advantage of the general relationship between least-squares solution and generalized inverses, the neural networks approach eliminates the need for inverting singular or poorly conditioned matrices and therefore also eliminates the need for the damping term often used to regularize such inversions. This procedure produces reconstructions with fewer artifacts and faster convergence than those attained previously using damped least-squares methods.