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The ARIANNA experiment is an Askaryan detector designed to record radio signals induced by neutrino interactions in the Antarctic ice. Because of the low neutrino flux at high energies (E-nu > 10(16 )eV), the physics output is limited by statistics. Hence, an increase in sensitivity significantly improves the interpretation of data and offers the ability to probe new parameter spaces. The amplitudes of the trigger threshold are limited by the rate of triggering on unavoidable thermal noise fluctuations. We present a real-time thermal noise rejection algorithm that enables the trigger thresholds to be lowered, which increases the sensitivity to neutrinos by up to a factor of two (depending on energy) compared to the current ARIANNA capabilities. A deep learning discriminator, based on a Convolutional Neural Network (CNN), is implemented to identify and remove thermal events in real time. We describe a CNN trained on MC data that runs on the current ARIANNA microcomputer and retains 95% of the neutrino signal at a thermal noise rejection factor of 10(5), compared to a template matching procedure which reaches only 10(2) for the same signal efficiency. Then the results are verified in a lab measurement by feeding in generated neutrino-like signal pulses and thermal noise directly into the ARIANNA data acquisition system. Lastly, the same CNN is used to classify cosmic-rays events to make sure they are not rejected. The network classified 102 out of 104 cosmic-ray events as signal.