Neural-Network-Based Variational Method in Nuclear Density Functional Theory: Application to the Extended Thomas-Fermi Model

We propose a neural-network-based variational framework for nuclear Density Functional Theory based on the extended Thomas–Fermi (ETF) model, in which proton and neutron number densities are represented by multilayer perceptrons and determined by direct minimization of a Skyrme-type energy density functional. We clarify the mathematical connection to the conventional Euler–Lagrange formulation, showing that stationarity in parameter space corresponds to a projected Euler–Lagrange condition on the neural-network trial-density manifold. The basic validity of the framework is examined through three sets of calculations: a Woods–Saxon potential benchmark, ground-state calculations of finite nuclei (${}^{40}$Ca, ${}^{90}$Zr, and ${}^{208}$Pb), and nuclear pasta phases. The binding energies of finite nuclei agree with existing ETF calculations to within $0.5\mathrm{\%}$, and representative pasta structures including spheres, rods, and slabs are reproduced. We also find that single-precision arithmetic yields results comparable to double precision, suggesting that the present framework is well suited to GPU environments in which low-precision computation is advantageous.