Reflection of photoacoustic (PA) signals from strong acoustic heterogeneities in biological tissue leads to reflection artifacts (RAs) in B-mode PA images. In practice, RAs often clutter clinically obtained PA images, making the interpretation of these images difficult in the presence of hypoechoic or anechoic biological structures. Towards PA artifact removal, several researchers have exploited 1) the frequency/spectrum content of time-series photoacoustic data in order to separate the true signal from artifacts, and 2) the multi-wavelength response of photoacoustic targets, assuming that the spectral nature of RAs correlates well with their corresponding source signals. These approaches are limited to extensive offline processing and sometimes fail to correctly identify artifacts in deep tissue. This study demonstrates the use of a deep neural network with the U-Net architecture to detect and reduce RAs in B-mode PA images. In order to train the proposed deep learning model for the RA reduction task, a program is designed to randomly generate anatomically realistic digital phantoms of human fingers with the capacity to produce RAs when subjected to PA imaging. In-silico PA imaging experiments, modeling photon transport and acoustic wave propagation, on these digital finger phantoms enabled the generation of 1800 training samples. The algorithm was tested on both PA images generated from digital phantoms and in-vivo PA data acquired from human fingers using a hand-held LED-based PA imaging system. Our results suggest that robust reduction of RAs with a deep neural network is possible if the network is trained with sufficiently realistic simulated images.