You cannot clear them, but you can create a new user and start the puzzles from scratch. On the title page, next to your name is a '...' button. If you touch that, you will go to a page where you can select which user you want to use, or create a new user by typing in a new name. You can have as many users as you want, and each will have their own set of puzzles so everyone in the family can play.
Note that you can also restart any puzzle that you want to replay without creating a new user.