Is it efficient to solve systems of differential equations using neural networks?

Question

Mathematics

If someone has come across the use of neural networks for solving systems of difurs in partial derivatives, can you tell me what are the advantages and disadvantages compared to classical methods, such as difference schemes and the finite element method? And in general, is it used at least somewhere in practice or not? There are scientific works on the topic, and quite old ones, for example, Vasiliev's dissertation from 2007, but I have not seen any implementations.
Subject area — gas dynamics, mechanics of a deformable solid body, magnetohydrodynamics. I am interested to understand whether it is worth doing this topic at all or not.

Answer 1

So that the question does not hang unanswered:
https://habr.com/ru/company/skillfactory/blog/526544/
https://arxiv.org/abs/2010.08895

Answer 2

After training, it is time to see what the neural network is capable of. Scientists tested it on 5000 equations, already without ready-made answers (none of them, however, could be classified as “unsolvable”). The neural network did an excellent job and found the right solutions for most of them. She was especially good at integration, solving almost 100% of these test problems, but did slightly worse at ordinary differentiation.

https://habr.com/ru/post/506722/
How to understand that a neural network will solve your problem. Great...

Answer 3

SINDy autoencoders .

Answer 4

https://arxiv.org/abs/1806.07366