Well, they weren’t purely for the sciences, but the idea I had as part of my economic development series was that, rather than having a system in which graduates of higher education were required to be polymaths (i.e, you need enough links on your chain to go around your neck and become a maester), you have schools set up where the objective is to train someone to competence in only one or two areas, and essentially create a new category of half-maester as credentialed specialists.
So for example, if you need more doctors, it makes sense to have schools where students go only to train to get silver links; or if you want to expand the communication network of the ravens out further, have people who only study to get black iron links; or if you need more accountants, have people who only study to get yellow gold links, and so on.
In terms of financing and location, my original idea was that a Lord Paramount would essentially subsidize the construction and operation of these schools by way of a payment to the Citadel, with schools located in cities/towns/major castles, in return for having their people educated at a reasonable price. And the target is essentially to build up the human capital of a given region, so that I have more literate people, more specialists in needed areas (not just doctors, but also lawyers, accountants, metallurgists, communications experts, teachers, etc.), and thus a more productive economy, although I imagine these halfmaester positions would probably be more attractive to the merchant, artisan, and wealthy peasant classes as a way to get their children a respected trade than to the nobility per se.