Self-consistent orbital-free nuclear density functional theory with a physics-constrained learned nonlocal kinetic energy functional

Nonlocal kinetic-energy density functionals (KEDFs) can encode nuclear shell structure in orbital-free density functional theory (OFDFT), but self-consistency requires accurate functional derivatives and a stable solution of the Euler equation. We construct a density-dependent-kernel KEDF whose correction is learned from Kohn-Sham (KS) reference data. Linearity in the kernel shape yields analytic Euler–Lagrange (EL) responses. The fit uses exact energy-matching equalities and soft quadratic inequality penalties for violations of prescribed tail-response and selected-path energy-rise margins, evaluated by an active-set repeated penalized least-squares iteration. We formulate the radial EL equation as the rearranged one-orbital eigenproblem. In a constant-$k_F$ ${}^{16}$O benchmark it converges without density mixing and agrees with imaginary-time evolution (ITE) to sub-keV energy. Adaptive-step ITE gives final species EL residuals of at most 0.07 MeV in the reported calculations, and rearranged diagonalization reaches the same stationary densities. For spherical $N=Z$ systems without spin–orbit or Coulomb terms, nucleus-specific fits for $A=16$ to $140$ reproduce shell patterns and radii. A correction trained on three nuclei transfers the shell pattern and radius, but not the absolute energy, to a previously unseen $A=140$ system.