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Synthetic aperture radar (SAR) images are inevitably contaminated by speckle noise due to its coherent imaging mechanism. Speckle noise obscures the intrinsic radar cross section (RCS) information in SAR images. This article proposes a novel deep neural network architecture specifically designed for despeckling purpose. It uses a convolutional neural network to extract image features and reconstruct a discrete RCS probability density function (PDF). It is trained by a hybrid loss function which measures the distance between the actual SAR image intensity PDF and the estimated one which is derived from convolution between the reconstructed RCS PDF and prior speckle PDF. The network can be trained by either purely simulated image patches or real SAR images. Experiment results on both simulated SAR images and real NASA/JPL AIRSAR images are used to test the performance, and the results show the efficacy of the proposed despeckling neural network compared with three state-of-the-art filters. [ABSTRACT FROM AUTHOR]