We apply recent advances in deep neural networks to three classes of geophysical problems stemming from ambient noise imaging: wavespeed inversion in homogeneous media in the presence of anisotropic sources, local wavespeed inversion in inhomogeneous media, and source directionality estimation in homogeneous media. Our networks are inspired by those commonly in the signal processing literature, such as convolutional networks and LSTMs, but use a training procedure that appears unique to physical problems: data is generated on the fly and only used to compute a single gradient, then discarded and never seen again. These techniques prove to be highly performant and quite flexible — they easily accomodate for data gathered from different sensor geometries, or for different priors in the data generating procedure. We also find preliminary evidence that, in simplified analogues of these problems, the nodes in deeper layers of our networks are computing physically meaningful quantities.