We present a study that uses machine learning to solve the forward and inverse scattering problems for synthetic aperture radar (SAR). Using a training set of known reflectivities as inputs and the resulting SAR measurements as outputs, the machine learning method produces an approximation for the sensing matrix of the forward scattering problem. Conversely, employing that same training set but with the SAR measurements used as inputs and the reflectivities as outputs, the machine learning method produces an approximate inverse of the sensing matrix. This learned approximate inverse mapping allows us to solve the inverse scattering problem as it maps SAR measurements to an estimate of the reflectivity. To interpret these results, we restrict our attention to a neural network arranged as a single, fully-connected layer. By doing so, we are able to interpret and evaluate the mappings produced by machine learning in addition to the results of those mappings. Employing a training set made up of 50,000 of the CIFAR-10 dataset as the reflectivities, we simulate SAR measurements using a physical model for the sensing matrix. With this training set of reflectivities and corresponding SAR measurements, we find that the results of machine learning accurately approximate the sensing matrix and provide a better answer to the inverse scattering problem than the standard SAR inversion formula. We also test the performance of the proposed methodology on a dataset with high resolution images while training with a lower resolution data set. The results are very promising showing again a superior performance for the learned approximate inverse mapping.