Is it possible to train a neural network to solve NP-complete problems?
To answer your question, I would to point you to the field of computational learning theory (CLT), which applies complexity theoretic approaches to analyse machine learning.
An important concept in CLT is probably approximately correct (PAC) learning: in simple terms, a problem is PAC learnable if there exists an efficient algorithm which learns the data using a polynomial number of samples from the underlying distribution of the problem with an (polynomially small) error of $\epsilon$ and (polynomially small) failure probability $\delta$.
Unfortunately, there is a big disconnect between results in CLT and results in applied machine learning, so you are unlikely to find result proving or disproving the learnability of NP complete problems using deep neural networks as it is still an active area to research.
Here are some resources to computational learning theory:

Wikipedia: https://en.wikipedia.org/wiki/Computational_learning_theory
The original paper on CLT by Leslie Valiant: http://web.mit.edu/6.435/www/Valiant84.pdf
The hardness of training a particular neural network is NP complete: https://people.csail.mit.edu/rivest/pubs/BR93.pdf
Very good set of notes on CLT:

https://www.cs.ox.ac.uk/people/varun.kanade/teaching/CLT-HT2018/lectures/lecture01.pdf
Section 3 defines and gradually introduces the concept of PAC learning

Section 5.1 of these notes show how 3-COLOURABILITY can be phrased in the PAC learning framework:
In Section 5 as well, the notes show how &quot;3-TERM-DNF is not efficiently PAC learnable (take II) unless RP = NP&quot;