Many computer-vision-related problems have successfully applied deep learning to improve the error rates with respect to classifying images. As opposed to optically based images, we have applied deep learning via a Siamese Neural Network (SNN) to classify synthetic aperture radar (SAR) images. This application of Automatic Target Recognition (ATR) utilizes an SNN made up of twin AlexNet-based Convolutional Neural Networks (CNNs). Using the processing power of GPUs, we trained the SNN with combinations of synthetic images on one twin and Moving and Stationary Target Automatic Recognition (MSTAR) measured images on a second twin. We trained the SNN with three target types (T-72, BMP2, and BTR-70) and have used a representative, synthetic model from each target to classify new SAR images. Even with a relatively small quantity of data (with respect to machine learning), we found that the SNN performed comparably to a CNN and had faster convergence. The results of processing showed the T-72s to be the easiest to identify, whereas the network sometimes mixed up the BMP2s and the BTR-70s. In addition we also incorporated two additional targets (M1 and M35) into the validation set. Without as much training (for example, one additional epoch) the SNN did not produce the same results as if all five targets had been trained over all the epochs. Nevertheless, an SNN represents a novel and beneficial approach to SAR ATR.