  We adopt the neural network flow (NN flow) method to study the
Berezinskii-Kosterlitz-Thouless (BKT) phase transitions of the 2-dimensional
q-state clock model with $q\ge 4$. The NN flow consists of a sequence of the
same units to proceed the flow. This unit is a variational autoencoder (VAE)
trained by the data of Monte-Carlo configurations in the way of unsupervised
learning. To gauge the difference among the ensembles of Monte-Carlo
configurations at different temperatures and the uniqueness of the ensemble of
NN-flowed states, we adopt the Jesen-Shannon divergence (JSD) as the
information-distance measure "thermometer". This JSD thermometer compares the
probability distribution functions of the mean magnetization of two ensembles
of states. Our results show that the NN flow will flow an arbitrary spin state
to some state in a fixed-point ensemble of states. The corresponding JSD of the
fixed-point ensemble takes a unique profile with peculiar features, which can
help to identify the critical temperatures of BKT phase transitions of the
underlying Monte-Carlo configurations.
