As promised, I went through the article as well as (call it Exhibit II upon which it builds -->)
There is so much that can be said, but I will reduce the scope by only looking at the :"numerics" (and not much in there). I did have a look some time ago and now again. It is really awful. I have experience of most of the numerical methods and in general, they don't work here or are not understood properly.
1. Exhibit II, section 2.1 are about 4 major issues.
a. fixed point iteration (even if you can find!) does not converge exponentially (it is linear, just try x = sin(x)). Robust contraction mappings are very difficult to construct.
b. Too many assumptions.
c. The jury is out on multilevel MC (does it work in theory?) and it is awful slow. (two theses).
d. A lot of symbolic formulae (sigm signs are always scary). ditto terms like "for sufficiently large/small N, h.".
Regarding your link
1. Strange PDEs (esoteric).
2. Seems Euler-Maruyama is at the heart of the work (this is not a good sign, it is a terrible method).
3. What is being approximated really? Table 1 is statistics, not numerics
, which uses [$]L\infty[$] error estimates.
4. Why not do a 1d and 2d PDE in detail for starters?
5. Performance? Not clear if the values in Table 2 are good or not.
I am unable to relate to much of the articles' contents. And the algorithms will break down.
On the other hand, maybe I am missing something really simple
Finally, to answer your specific questions:
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?
Yes, quite a few, above.
Finally, who on earth solved a PDE by posing it as an optimal control/.BSDE/DL? If you are serious show for 1 factor early exercise option as Proof of Concept.