And, guess what? Since, the Galerkin approximation requires one to choose a basis that is appropriate to the problem at hand, we now have a deep learning solution too (since neural network learning is essentially equivalent to learning an adaptive basis). It is called the Deep Galerkin Method [1]. In a nutshell, the method directly minimizes the L2 error over the PDE, boundary conditions and initial conditions. The i…
Why would you use the Monte Carlo method when the quasi-Monte Carlo method converges so much more quickly? I admit, I am a little biased, because I worked on some QMC stuff in grad school, but it works really, really well in practice. https://en.wikipedia.org/wiki/Quasi-Monte_Carlo_method
If the number of dimensions is high enough then pretty much all points are far away from each other so there's no real need to worry about points clustering. You're at a much greater risk of simply missing interesting parts of your space than you are of oversampling any part of it.