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Using a supervised neural network (NN) trained once on a one-dimensionallattice of 200 sites, we calculate the Berezinskii--Kosterlitz--Thouless phasetransitions of the two-dimensional (2D) classical $XY$ and the 2D generalizedclassical $XY$ models. In particular, both the bulk quantities Binder ratiosand the spin states of the studied systems are employed to construct the neededconfigurations for the NN prediction. By applying semiempirical finite-sizescaling to the relevant data, the critical points obtained by the NN approachagree well with the known results established in the literature. This impliesthat for each of the considered models, the determination of its various phasesrequires only a little information. The outcomes presented here demonstrateconvincingly that the employed universal NN is not only valid for the symmetrybreaking related phase transitions, but also works for calculating the criticalpoints of the phase transitions associated with topology. The efficiency of theused NN in the computation is examined by carrying out several detailedbenchmark calculations.

Berezinskii--Kosterlitz--Thouless transition -- a universal neural network study with benchmarking