The ARIANNA experiment is a detector designed to record radio signals created by high-energy neutrino interactions in the Antarctic ice. Because of the low neutrino rate at high energies, the physics output is limited by statistics. Hence, an increase in detector sensitivity significantly improves the interpretation of data and offers the ability to probe new physics. The trigger thresholds of the detector are limited by the rate of triggering on unavoidable noise. A real-time noise rejection algorithm enables the thresholds to be lowered substantially and increases the sensitivity of the detector by up to a factor of two compared to the current ARIANNA capabilities. Deep learning discriminators based on Fully Connected Neural Networks (FCNN) and Convolutional Neural Networks (CNN) are evaluated for their ability to reject a high percentage of noise events (while retaining most of the neutrino signal) and to classify events quickly. In particular, we describe a CNN trained on Monte Carlo data that runs on the current ARIANNA microcontroller and retains 95% of the neutrino signal at a noise rejection factor of 10^5.