It's been quite a while but I have some very fuzzy memories of a lab I worked with in undergrad working on some sort of optimistic methods for solving PDEs, something along the lines of speculatively projecting into a lower dimension space where you can tell if the projection has become inaccurate.

This was well before neural nets were hot and the term 'machine learning' was fresh on the horizon. I believe the problem largely revolved around discovering the best set of basis functions to project onto.

I don't remember if there were provably 'best' projections but if not, there might be something interesting there