Name of classification/trope of anime with advanced/ancient civilizations I would like to use "Laputa: Castle in the Sky" as an example, but the identifying features that I'm particularly interested in identifying are the ancient, yet advanced technological civilization(s) that left behind the city itself.

Is there a word or genre specifically describing the ancient yet technologically advanced civilization trope (specifically those more advanced than present day civilizations)?
"Precursors" or "forerunners" would be the title given to the society/societies that came before but they are not titles that would be classified as a genre.   The closest thing to an official genre title that you would get is, "post-apocalyptic". Post-apocalyptic settings are those that take place after the end of the world. in certain instances the end of the world does not mean the end of humanity and so they start the world anew. I would recommend searching post-apocalyptic animes if reborn society is what you are into. Attack On Titan, Gargantia on the Verdurous Planet, and Murder Princess, are all examples of animes with functional societies that speak of "technology from the old world", even though the ancient technology is futuristic by comparison to their own. Enjoy