All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad is His slave and Messenger.
Before answering your question, we would like to inform you that the Muslim scholars disagreed on whether or not one’s fast becomes invalid if it is forgetfully broken. According to the view of the majority, fasting is not invalidated, and this is the predominant opinion. Once he remembers, the fasting person is only required to resume his fast for the rest of the day.
As per your question, according to the majority of the scholars, the expiation for breaking the fast is not due unless the fasting is broken on account of sexual intercourse. According to the Maliki scholars, expiation is due on the fasting person if he breaks his fast intentionally. Yet they stipulate that the fasting person breaks his fast with the deliberate intention of violating the sacredness of the month.
In At-Taj and Al-Ekleel, Ibn ‘Arafah says, “The expiation is due for breaking the fast in Ramadan with the deliberate intention of violating the sacredness of the month.” Somewhere else, the book quotes Malik as saying in Al-Mudawwanah:
If someone has not made the intention of fasting for one day in Ramadan thinking that the day was not part of Ramadan, and he later realizes that it is Ramadan and this is before he eats or drinks, or let us suppose that he does not realize the day is Ramadan until after he has already eaten or drunk, then once he realizes this, he is required to abstain from eating or drinking for the rest of the day. If he continues eating or drinking after he has realized that the day was Ramadan, then he is required to make up that day later, and no expiation is due on him unless he continues breaking the fast with the deliberate intention of violating the sacredness of the month and while being aware of the prohibition of deliberate breaking the fast in Ramadan.
In conclusion, if someone breaks his fast in Ramadan out of forgetfulness and he does not resume fasting during the rest of the day after remembering it because he is ignorant of the fact that he has to, then he is only required to make up that day later.
Allah knows best.