No. He didn't find out until he read Luke's thoughts and feelings in ROTJ. Her Force affinity was heavily buried within her compared to Luke, so that was one of many reasons Vader didn't sense it from her in the first two films. He truly believed she was someone else's daughter.
Based on what little the EU revealed, the official story was that Padmé died with her "baby" still inside her, and the "child" died along with her and both were buried in state on Naboo. (No doubt Vader bought this story and believed it too, considering what he almost did to his own wife). There was a minor incident where Vader secretly went to visit his wife's tomb and was driven away by a secret group of female warriors whose job was to guard the royal tombs of Naboo. (I have a feeling he could have taken them, but he didn't want to draw attention to what he was doing, so he chose to leave quietly).
Vader didn't know anything about his wife giving birth before she died, nor that she was pregnant with fraternal twins, or that they survived and were taken away to different worlds and adopted. It wouldn't surprise me if Obi-Wan and/or Yoda even performed a force ritual that shielded both kids from Vader's powers for a brief time while they were growing up.
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