My husband is a heavy stirrer and our everyday porcelain dishes were showing flatware scuffing. I tried the Weiman's Stove Top Cleaner. Rubbing with a damp paper towel, this is the cleaning results from a moderate swiping motion. I did half of the bowl for comparison purposes.
I use Bar Keepers Friend for that same purpose. It probably works the same way and doesn't hurt the surface at all.