In many areas of data science, Deep Neural Networks have exhibited a remarkable ability to learn complex, non-linear relationships between sets of variables. In this paper, we apply this network architecture to the task of stochastic estimation, first proposed by Adrian over 40 years ago. Two Deep Neural Networks (DNNs) are trained to estimate the pressure at selected locations in azimuthal and stream-wise arrays of pressure transducers situated just outside a Mach 0.6 jet, given instantaneous pressure measurements made at other locations in the arrays. The estimated pressure is compared with the instantaneous pressure fluctuations measured at each of the selected locations, as well as the estimates made using traditional Linear Stochastic Estimation (LSE). The root-mean-square error between the values predicted by the DNN and those measured by the transducers is shown to be nearly identical to the error associated with the traditional LSE method in the case of the azimuthal array. For the stream-wise configuration, the DNN shows an approximately 60% reduction in error over the LSE model. In addition, some limitations and possible extensions of this method are discussed.