The rapid localization of radioactive material in an urban environment is of critical importance to secure radiological sources and prevent radiological attacks. We consider the inverse problem of inferring the three-dimensional location of stationary and moving radiation sources given a set of measurements from an array of radiation sensors. A feedforward neural network is employed to quickly infer the location of the radioactive source. We optimize the weights of the neural network using Nadam gradient-based optimization. This method of source localization lacks the prediction intervals given by other techniques, such as Bayesian inference, but it is extremely fast, so it enables real-time predictions. We utilize this advantage to track the position of a moving radioactive source within a simulated urban environment.