Here's an idea I had while working on google's HEIR project as an open source contributor.

Essentially you're using a wreath product to encode a problem of the form:

"At each position p, select a transformation from group G where positions can be permuted by group H"

For cases where the cohomology doesn't vanish to something trivial, I use sheaves to "glue" local solutions together to for a more global solution around any obstructions.

I found this cool because it's a non iterative (straight least-squares) method that was able to perform a multitude of tasks (counting, indexing, fluid dynamics simulation, and simple proof generation).

Would love to get any feedback on the model. My gh heir fork has an experimental branch with the implementation if anyone is interested in that.