Cuchulainn wrote:It's a very ambitious article..
But what about its correctness? Does it look like it has solved a problem with, say, more accuracy and speed than the state of the art classical numerical methods? Do you see any flaws like you saw in the paper you discussed in the thread quoted below?
Cuchulainn wrote:ISayMoo wrote:Cuchulainn wrote:Your call on the math being outdated and fairly basic is missing the point a bit I think.
I may have missed a point, especially since you didn't give one. All I see is a link to a not useful article (that I waded in, not even a hint to pinpoint what's on your mind) and DL/PDE buzzwords in the title.
What's the question centering around DL/PDE,exactly? A guess; does DL solve curse of dimensionality?
// is there a GOOD paper on DL/PDE. 1st impression is it's a solution looking for a problem. 'Teaching' a pde sounds a bit weird.
I think this one is a bit better: https://arxiv.org/abs/1708.07469 Yes, they argue that DL solves the curse of dimensionality.
When I first saw this "note' I thought it looked good but on deeper inspection it turned out to be a train wreck/horrror show. Part of my research back then was proving convergence of FEM (finite element) in Sobolev spaces, later in TV design, oil and gas. So, It is very interesting to see how someone uses maths maths to meet DL half way.
The background of the authors is DL and business (nothing wrong with that) but the article is riddled with errors that I don't even know where to begin:
1. The Galerkin method fell into disuse around 1943. More seriously, the article is really the MESHLESS method which has its own issues.
2. It is not even DL imo; just because you use a stochastic gradient method does not make it as DL.
3. "DGM is a natural merging of ML and Galerkin". Yes?
4.Details are missing (e.g. Table 1!!!!!!).
5. I seems DL applications need some kind of analytic solution for training? PDEs don't have this in general.
6. "Approximate the second derivatives using Monte Carlo". This is fiction.
7. Major numerical difficulties can't be swept under the carpet. In fact, DL will compound them, e.g how to choose an optimal learning rate alpha (usually use Brent's method).
I don't see much synergy between DL and PDE at this stage.
- that is really only the known knowns and the known unknowns. And each year, we discover a few more of those unknown unknowns.
AFAIR Meshless leads to a full (ill-conditioned) matrix system