  High-amplitude events of the out-of-plane vorticity component $\omega_z$ are
analyzed by stereoscopic particle image velocimetry (PIV) in the bulk region of
turbulent Rayleigh-B\'{e}nard convection in air. The Rayleigh numbers ${\rm
Ra}$ vary from $1.7 \times 10^4$ to $5.1 \times 10^5$. The experimental
investigation is connected with a comprehensive statistical analysis of
long-term time series of $\omega_z$ and individual velocity derivatives
$\partial u_i/\partial x_j$. A statistical convergence for derivative moments
up to an order of 6 is demonstrated. Our results are found to agree well with
existing high-resolution direct numerical simulation data in the same range of
parameters, including the extreme vorticity events which appear in the far
exponential tails of the corresponding probability density functions. The
transition from a Gaussian to a non-Gaussian velocity derivative statistics in
the bulk of a convection flow is confirmed experimentally. The experimental
data are used to train a reservoir computing model, one implementation of a
recurrent neural network, to reproduce highly intermittent experimental time
series of the vorticity and thus reconstruct extreme out-of-plane vorticity
events. After training the model with high-resolution PIV data, the machine
learning model is run with sparsely seeded, continually available, and unseen
measurement data in the reconstruction phase. The dependence of the
reconstruction quality on the sparsity of the partial observations is also
documented. Our latter result paves the way to machine--learning--assisted
experimental analyses of small-scale turbulence for which time series of
missing velocity derivatives can be provided by generative algorithms.
