Traveling Salesman Problem with Neural Network
there are many papers on using artificial neural networks to solve TSP including recurrent and Hopfield networks, and they "succeed" in a rough sense, but so far there does not seem to be any evidence that the techniques are in any way (strongly?) superior to other algorithmic approaches, so its something more like a research curiosity at the moment. the use of ANNs for this problem is indeed counterintuitive from the pov of combinatorial algorithmics and the mechanisms by which the problem inputs/ outputs are encoded are novel and tend to vary, and maybe are not yet so standardized. the authors seem maybe more interested in demonstrating "proof of concept" and a comparison with other algorithm types seems more rare (there is some in the last paper). see eg


Traveling salesman problem using neural network techniques / Abdel-Moetty
Using Hopfield Networks to Solve Traveling Salesman Problems Based on Stable State
Analysis Technique / Feng, Douligeris
Comparison of Neural Networks for Solving the Travelling Salesman Problem / La Maire, Mladenov
A Recurrent Neural Network to Traveling
Salesman Problem / Siqueira, Scheer, Steiner