  Using the technique of supervised neural networks (NN), we study the phase
transitions of two-dimensional (2D) 6- and 8-state clock models on the square
lattice. The employed NN has only one input layer, one hidden layer of 2
neurons, and one output layer. In addition, the NN is trained without any prior
information about the considered models. Interestingly, despite its simple
architecture, the built supervised NN not only detects both the two
Berezinskii--Kosterlitz--Thouless (BKT) transitions but also determines the
transition temperatures with reasonable high accuracy. It is remarkable that a
NN, which has an extremely simple structure and is trained without any input
from the studied models, can be employed to study topological phase
transitions. The outcomes shown here as well as those previously demonstrated
in the literature suggest the feasibility of constructing a universal NN that
is applicable to investigate the phase transitions of many systems.
