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Millimeter-wave communication has the potential to deliver orders of magnitude increases in mobile data rates. A key design challenge is to enable rapid beam alignment with phased arrays. Traditional millimeter-wave systems require a high beam alignment overhead, typically an exhaustive beam sweep, to find the beam direction with the highest beamforming gain. Compressive sensing is a promising framework to accelerate beam alignment. However, model mismatch from practical array hardware impairments poses a challenge to its implementation. In this work, we introduce a neural network assisted compressive beam alignment method that uses noncoherent received signal strength measured by a small number of pseudorandom sounding beams to infer the optimal beam steering direction. We experimentally showcase our proposed approach with a 60GHz 36-element phased array in a suburban line-of-sight environment. The results show that our approach achieves post alignment beamforming gain within 1dB margin compared to an exhaustive search with 90.2 percent overhead reduction. Compared to purely model-based noncoherent compressive beam alignment, our method has 75 percent overhead reduction.