For practical purposes, functionals are just infinite-dimensional functions. In practice, a function is always represented numerically using a finite basis (grid, splines, Fourier), so functionals become just high-dimensional functions. At which point we are back to ordinary machine learning. So I’m not sure what’s the point here.
Exactly, I'm not sure why we need a new fancy name for this obvious task.
How well does this scale? Say I have a discrete Poisson equation for some 3d geometry [1], how does the solution time compare to fast multigrid methods, for increasing matrix size?