There is conditioning you can do to strengthen your arch. Whether this will change the shape of your foot significantly, or "cure" you of your flat feet, I don't know.
Here are a few exercises you may want to try:
1. Stand on a small towel that is laid out flat on the floor. Use your toes to grab and scrunch the towel under your foot bit by bit. Repeat until the towel is all scrunched under your foot. Then use your toes to gradually push the towel back out flat again.
2. While sitting, flex and point the feet. Be sure when you point that you are pushing the top of your ankle towards the ceiling, pulling your heel high towards your calf, and reaching your toes long, so that you are pointing the whole foot and not just "clawing" the toes. Can also be done with a Theraband for more resistance.
3. Rise up onto the balls of your feet, as high as you can, and then lower the heels slowly.
4. Sitting in a chair with feet together, lift the heels as high as you can. Then, keeping the ankle stable, use just the toes to "push the floor away from you" (you will actually go onto tiptoe). Relax the toes, but keep the heels high and the ankle and arch solid. (Your foot at this point should look like a Barbie doll foot... pre-formed so she can put her high heels on, you know?)

Keep repeating the action in the toes for several reps.
Hope these help.