An inversion technique is presented for retrieving vertical profiles of atmospheric temperature and vapor from the brightness temperatures measured by a ground-based multichannel microwave radiometer and the surface measurements of temperature and relative humidity. It combines a profile expansion over a complete set of natural orthogonal functions with a neural network which performs the estimate of the coefficients of the expansion itself. A simulation study has been carried out, and the algorithm has been tested by comparing its retrievals with those obtained by means of linear statistical inversion applied on the same data sets. The analysis has been limited to the case of profiles with clouds in order to test the ability of the neural network to face nonlinear problems. The technique has proven to be flexible, showing a good capability of exploiting information provided by other instruments, such as a laser ceilometer. A fault tolerance evaluation has also been considered, which showed interesting properties of robustness of the algorithm.