Machine Learning applied to database design
This is such an interesting question. I suppose that it is possible but you would have to answer some more questions before you can actually get help with modeling something. 


Are you looking for it to learn SQL or NoSQL? 
You'd have to make a distinction between something that can learn relational database design versus something that learns how to be a DBA and work in a particular language. For example, relational databases are based on theory (and relatively straightforward) but how you implement certain things in Oracle or SQL Server (as examples) will vary greatly. Or maybe you're looking for a particular type of design like data warehousing (star patterns, etc). Whichever approach you choose would have a profound effect on the type of model you are going to build.
There are some pitfalls that you would have to account for. A relationship based on text columns is acceptable design, but a relationship based on integer hashes of those same text columns is much better. How a model would account for something like this is unknown to me. 
Relating to the item above, you would have to come up with some metric for the success of your model. Is it the levels of relational design that it can reach? Is it some hardware performance benchmark? Is it some level of cognition that your model can reach for extremely complex designs?


I think that once you answer these types of questions you will be in a much better position to start model development.