Radio frequency (RF) fingerprinting is a tool which allows for authentication by utilizing distinct and random distortions in a received signal based on characteristics of the transmitter. We introduce a deep learning-based authentication method for a novel RF fingerprinting system called Physically Unclonable Wireless Systems (PUWS). An element of PUWS is based on the concept of Chaotic Antenna Arrays (CAAs) that can be cost effectively manufactured by utilizing mask-free laser-enhanced direct print additive manufacturing (LE-DPAM). In our experiments, using simulation data of 300 CAAs each exhibiting 4 antenna elements, we test 3 different convolutional neural network (CNN) architectures under different channel conditions and compare their authentication performance to the current state-of-the-art RF fingerprinting authentication methods.