Hunting for open clusters in Gaia EDR3: $628$ new open clusters found with OCfinder

The improvements in the precision of the published data in \textit{Gaia} EDR3 with respect to \textit{Gaia} DR2, particularly for parallaxes and proper motions, offer the opportunity to increase the number of known open clusters in the Milky Way by detecting farther and fainter objects that have so far go unnoticed. Our aim is to keep completing the open cluster census in the Milky Way with the detection of new stellar groups in the Galactic disc. We use \textit{Gaia} EDR3 up to magnitude $G = 18$ mag, increasing in one unit the magnitude limit and therefore the search volume explored in our previous studies. We use the \texttt{OCfinder} method to search for new open clusters in \textit{Gaia} EDR3 using a Big Data environment. As a first step, \texttt{OCfinder} identifies stellar statistical overdensities in the five dimensional astrometric space (position, parallax and proper motions) using the \texttt{DBSCAN} clustering algorithm. Then, these overdensities are classified into random statistical overdensities or real physical open clusters using a deep artificial neural network trained on well-characterised $G$, $G_{\rm BP} - G_{\rm RP}$ colour-magnitude diagrams. We report the discovery of $664$ new open clusters within the Galactic disc, most of them located beyond $1$ kpc from the Sun. From the estimation of ages, distances and line-of-sight extinctions of these open clusters, we see that young clusters align following the Galactic spiral arms while older ones are dispersed in the Galactic disc. Furthermore, we find that most open clusters are located at low Galactic altitudes with the exception of a few groups older than $1$ Gyr. We show the success of the \texttt{OCfinder} method leading to the discovery of a total of $1\,310$ open clusters (joining the discoveries here with the previous ones based on \textit{Gaia} DR2)[abridged]