>Finite Difference method
>Finite Element method
>Finite Volume method

What's the next big mathematical modelling / PDE solving paradigm? I saw a paper talking about a grid-free approach using Monte Carlo methods earlier this week. What other avenues could be pursued? What about approaching the solution to a PDE over some geometry as a (constrained) optimisation problem? Some kind of "minimise LHS minus RHS" or something? Is there anything we can use from machine learning? (which is basically just an optimisation problem anyways)

Let's discuss