  Predictions for the two-photon exchange (TPE) correction to unpolarized $ep$
elastic cross section, obtained within two different approaches, are confronted
and discussed in detail. In the first one the TPE correction is extracted from
experimental data by applying the Bayesian neural network (BNN) statistical
framework. In the other the TPE is given by box diagrams, with the nucleon and
the $P_{33}$ resonance as the hadronic intermediate states. Two different form
factor parametrizations for both the proton and the $P_{33}$ resonance are
taken into consideration. Proton form factors are obtained from the global fit
of the full model (with the TPE correction) to the unpolarized cross section
data. Predictions of both methods agree well in the intermediate $Q^2$ range,
$(1,3)$ GeV$^2$. Above $Q^2=3$ GeV$^2$ the agreement is on $2\sigma$ level.
Below $Q^2=1$ GeV$^2$ the consistency between both approaches is broken. The
values of the proton radius extracted within both models are given. In both
cases predictions for VEPP-3 experiment have been obtained and confronted with
the preliminary experimental results.
