Understanding Machine-learned Density Functionals
This work addresses the challenge of developing machine-learned density functional approximations for quantum systems, which is incremental but domain-specific.
The researchers tackled the problem of approximating kinetic energy as a functional of density for non-interacting fermions in a 1D box using kernel ridge regression, achieving highly accurate energies and constrained optimal densities through modified Euler-Lagrange minimization and a projected gradient descent algorithm.
Kernel ridge regression is used to approximate the kinetic energy of non-interacting fermions in a one-dimensional box as a functional of their density. The properties of different kernels and methods of cross-validation are explored, and highly accurate energies are achieved. Accurate {\em constrained optimal densities} are found via a modified Euler-Lagrange constrained minimization of the total energy. A projected gradient descent algorithm is derived using local principal component analysis. Additionally, a sparse grid representation of the density can be used without degrading the performance of the methods. The implications for machine-learned density functional approximations are discussed.