Also having the same problem with my physical copy. I've gotten as far as 60% before it throws out the "Disk cannot be read. Make sure the disk is not damaged or dirty." message, but it has also happened at as early as 3%. Most of time it's around the half-way mark. I've also had an error saying that the application data is corrupt, and to delete the data and reinstall. I've tried rebuilding the database, deleting and reinstalling, waiting for the patch, ignoring the patch, and just restarting the console several times with no luck. I also tried installing a few others games to make sure it wasn't my PS4 - all the other games installed without any problems.
Sometimes, when I 'Close Application and Remove Disk' like the error pop up says, then put the disk back in and retry, the instillation will resume from where it left off, but other times it will restart from zero. It just crashes again within a minute, regardless. I have to wonder, though, if it's still actually installing. Even after the error message pops up I can hear the SPECIAL videos playing in the background, taunting me.