The A320 series don't weigh themselves on the ground. But once you are airborne it can use information from the Angle of Attack probes to work out its actual weight. It uses this to calculate such characteristic speeds for each flap setting. It displays the relevant speeds on the Primary Flight Display and so if you wish you can make the reverse calculation to work out the actual weight the aircraft thinks it is.
There is a formula in FCOM 3.04.10 and also the QRH has a table in 4.01