Buckingham pi theorem is bad?

The Buckingham Pi theorem is about making physics problems dimensionless. It says that if you have a law of physics that you can manipulate into the form f(inputs) = 0, you can re-write that law with fewer, dimensionless inputs. It's interesting, and important, and it motivated the work that Soledad Villar (JHU) and I are doing on making machine-learning methods obey exact dimensional scalings and unit conversion symmetries.

However, I am not sure the Buckingham Pi theorem works (or is useful) when the function f() is a vector-valued or tensor-valued function with vector-valued and tensor-valued inputs, as it is, say in Maxwell's equations or the equations of general relativity. Villar and I discussed ways to save Buckingham Pi, but I think the main results might either not be correct at all, or not reduce dimensionality. I got upset about it! But it raises an interesting question: Can Buckingham Pi be saved?

My point is: Physics is full of vectors and tensors and the laws are coordinate-free. If going dimensionless is a good idea, then it should be a good idea for vector and tensor expressions that are coordinate-free!