  In principle, many-electron correlation energy can be precisely computed from
a reduced Wigner distribution function ($\mathcal{W}$) thanks to a universal
functional transformation ($\mathcal{F}$), whose formal existence is akin to
that of the exchange-correlation functional in density functional theory. While
the exact dependence of $\mathcal{F}$ on $\mathcal{W}$ is unknown, a few
approximate parametric models have been proposed in the past. Here, for a
dataset of 923 one-dimensional external potentials with two interacting
electrons, we apply machine learning to model $\mathcal{F}$ within the kernel
Ansatz. We deal with over-fitting of the kernel to a specific region of
phase-space by a one-step regularization not depending on any hyperparameters.
Reference correlation energies have been computed by performing exact and
Hartree--Fock calculations using discrete variable representation. The
resulting models require $\mathcal{W}$ calculated at the Hartree--Fock level as
input while yielding monotonous decay in the predicted correlation energies of
new molecules reaching sub-chemical accuracy with training.
