Although many near-Earth objects have been found by ground-based telescopes, some fast-moving ones, especially those near detection limits, have been missed by observatories. We developed a convolutional neural network for detecting faint fast-moving near-Earth objects. It was trained with artificial streaks generated from simulations and was able to find these asteroid streaks with an accuracy of 98.7% and a false positive rate of 0.02% on simulated data. This program was used to search image data from the Zwicky Transient Facility (ZTF) in four nights in 2019, and it identified six previously undiscovered asteroids. The visual magnitudes of our detections range from ~19.0 - 20.3 and motion rates range from ~6.8 - 24 deg/day, which is very faint compared to other ZTF detections moving at similar motion rates. Our asteroids are also ~1 - 51 m diameter in size and ~5 - 60 lunar distances away at close approach, assuming their albedo values follow the albedo distribution function of known asteroids. The use of a purely simulated dataset to train our model enables the program to gain sensitivity in detecting faint and fast-moving objects while still being able to recover nearly all discoveries made by previously designed neural networks which used real detections to train neural networks. Our approach can be adopted by any observatory for detecting fast-moving asteroid streaks.