  The Berezinskii--Kosterlitz--Thouless (BKT) transition of the two-dimensional
$XY$ model on the honeycomb lattice is investigated using both the techniques
of Neural Network (NN) and Monte Carlo simulations. It is demonstrated in the
literature that with certain plausible assumptions, the associated critical
temperature $T_{\text{BKT,H}}$ is found to be $\frac{1}{\sqrt{2}}$ exactly.
Surprisingly, the value of $T_{\text{BKT,H}}$ obtained from our NN calculations
is 0.560(9) which deviates significantly from $\frac{1}{\sqrt{2}}$. In
addition, based on the helicity modulus, the $T_{\text{BKT,H}}$ determined is
0.571(8) agreeing well with that resulting from the NN estimation. The outcomes
presented in this study indicate that a detailed analytic calculation is
desirable to solve the found discrepancy.
