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Deep learning has been recently shown to improve performance in the domain of synthetic aperture sonar (SAS) image classification. Given the constant resolution with range of a SAS, it is no surprise that deep learning techniques perform so well. Despite deep learning's recent success, there are still compelling open challenges in reducing the high false alarm rate and enabling success when training imagery is limited, which is a practical challenge that distinguishes the SAS classification problem from standard image classification set-ups where training imagery may be abundant. We address these challenges by exploiting prior knowledge that humans use to grasp the scene. These include unconscious elimination of the image speckle and localization of objects in the scene. We introduce a new deep learning architecture which incorporates these priors with the goal of improving automatic target recognition (ATR) from SAS imagery. Our proposal -- called SPDRDL, Structural Prior Driven Regularized Deep Learning -- incorporates the previously mentioned priors in a multi-task convolutional neural network (CNN) and requires no additional training data when compared to traditional SAS ATR methods. Two structural priors are enforced via regularization terms in the learning of the network: (1) structural similarity prior -- enhanced imagery (often through despeckling) aids human interpretation and is semantically similar to the original imagery and (2) structural scene context priors -- learned features ideally encapsulate target centering information; hence learning may be enhanced via a regularization that encourages fidelity against known ground truth target shifts (relative target position from scene center). Experiments on a challenging real-world dataset reveal that SPDRDL outperforms state-of-the-art deep learning and other competing methods for SAS image classification.