Objective and subjective quality assessment is still a challenging problem in various image processing tasks. For instance, in the context of image compression, most of the conducted studies have focused on image datasets encoded using standard algorithms such as JPEG and JPEG2000. In this paper, we propose to further investigate the quality assessment issue in the presence of neural networks-based compressed images. More precisely, a new database of compressed images has been firstly built using JPEG2000 standard as well as four recent neural networks based coding schemes. Then, subjective experiments are performed to obtain the mean opinion scores of the generated distorted images. Finally, an extensive evaluation and analysis of objective image quality assessment metrics is achieved. For instance, in addition to conventional and machine learning metrics, we have considered different deep learning based models, which have been trained on our database. The new subjective database with its associated mean opinion scores as well as the learned models are publicly available at https://github.com/zakopz/NNCD-IQA-Database. The obtained results show the interest of deep learning based metrics in the context of neural networks-based compressed images.