Is a marriage contracted without witnesses religiously valid, especially given the presence of children, and what measures should be taken concerning this?

This marriage is invalid according to the majority of scholars, based on the hadith (definition): "There is no marriage except with a guardian and two just witnesses." However, Al-Zuhri, Abu Thawr, and Ibn Al-Mundhir hold the view that marriage is valid without witnesses, and this is also narrated from Imam Ahmad. Nevertheless, the opinion that requires witnesses is the preponderant one. The man must enter into a valid contract with her, fulfilling all the conditions. As for the children, they are attributed to him and inherit from him because they are the product of intercourse performed under a misconception [waty shubha (definition)].