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 $0.5\%$, 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.