Is a man's marriage to a woman with whom he committed adultery invalid, and must he divorce her, despite their repentance and having children together?

As long as you both repented from adultery before marriage, your marriage is valid and there is no problem with it, for the Prophet, peace and blessings be upon him, said: "The one who repents from sin is like one who has no sin." Repentance is by seeking forgiveness, feeling remorse, and refraining from the sin. You must keep your wife and treat her kindly, and strive in obedience to Allah and thanking Him for granting you success in repentance and a good marital life.