A neural network approach to the inverse scattering problem for microwave tomographic reconstruction is presented. Although the technique of microwave tomography has been developed for more than two decades, it is still in its infancy in that it is necessary to solve the inverse scattering problem, which is well known as an ill-posed and nonlinear problem and therefore difficult to deal with. To improve the inherent ill-posedness of the problem, good regularization procedures are required. In this paper, an edge-preserving regularization is proposed with a set of line processes to preserve the edge of the reconstructed image. Since the unknown dielectric permittivities are continuous complex variables and the line processes are binary variables, an augmented Hopfield network is applied to the mixed-variable optimization problem. With this method, a priori knowledge can be conveniently incorporated into the optimization process, and inversions of large matrices are avoided. A numerical example of a simple model illuminated by the transverse magnetic incident waves is reported, and the advantages and limitations of the method are discussed.