Flat Feet (also called collapsed arches) are a difficult problem to fix. Luckily, many people who think they have flat feet actually just have collapsed inner ankles. That's simpler to fix because the ankles are more flexible and reactive to training.
If your arches really are collapsed, however, you'll need to do the following exercises to reestablish the feeling of arching the foot. By training in this passive way you can reactivate the tendons and ligaments in the feet and hopefully add a little bit of the arch back, one step at a time.