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In principle, many-electron correlation energy can be precisely computed froma reduced Wigner distribution function ($\mathcal{W}$) thanks to a universalfunctional transformation ($\mathcal{F}$), whose formal existence is akin tothat of the exchange-correlation functional in density functional theory. Whilethe exact dependence of $\mathcal{F}$ on $\mathcal{W}$ is unknown, a fewapproximate parametric models have been proposed in the past. Here, for adataset of 923 one-dimensional external potentials with two interactingelectrons, we apply machine learning to model $\mathcal{F}$ within the kernelAnsatz. We deal with over-fitting of the kernel to a specific region ofphase-space by a one-step regularization not depending on any hyperparameters.Reference correlation energies have been computed by performing exact andHartree--Fock calculations using discrete variable representation. Theresulting models require $\mathcal{W}$ calculated at the Hartree--Fock level asinput while yielding monotonous decay in the predicted correlation energies ofnew molecules reaching sub-chemical accuracy with training.

Machine Learning Modeling of Wigner Intracule Functionals for Two Electrons in One Dimension