Artificial neural networks for density-functional optimizations in fermionic systems

Abstract

In this work we propose an artificial neural network functional to the ground-state energy of fermionic interacting particles in homogeneous chains described by the Hubbard model. Our neural network functional was proven to have an excellent performance: it deviates from numerically exact calculations by less than 0.15% for a vast regime of interactions and for all the regimes of filling factors and magnetizations. When compared to analytical functionals, the neural functional was found to be more precise for all the regimes of parameters, being particularly superior at the weakly interacting regime: where the analytical parametrization fails the most, ~7%, against only ~0.1% for the neural network. We have also applied our homogeneous functional to finite, localized impurities and harmonically confined systems within density-functional theory (DFT) methods. The results show that while our artificial neural network approach is substantially more accurate than other equivalently simple and fast DFT treatments, it has similar performance than more costly DFT calculations and other independent many-body calculations, at a fraction of the computational cost.

Machine learning for the solution of the Schrödinger equation

“…A multi-layer NN was also used to learn the electron correlation in the Anderson-Hubbard model by Ma et al [96]. Custodio et al [97] used single-and two-hidden layer NN to learn an LSDA type functional for a one-dimensional Hubbard model. Schmidt et al [98] used a neural network to learn the universal exchange-correlation functional that simultaneously reproduces both the exact exchange-correlation energy and the potential, for a one-dimensional system with two strongly correlated electrons, in a non-singular potential.…”

Electron-electron versus electron-phonon interactions in lattice models: Screening effects described by a density functional theory approach

“…In addition to ab initio descriptions of matter, (TD)DFT has been applied to model Hamiltonians to explore conceptual and methodological aspects of the theory [39][40][41][42][43][44][45][46][47][48][49][50][51], as well as for specific applications to cold atoms [52][53][54][55], Kondo physics [56][57][58], quantum transport [59][60][61], quantum electrodynamics [33], and nonequilibrium thermodynamics [62], to mention a few [63]. We here consider a two-component DFT for the HH model, where the basic variables are given by the set (n, x) ≡ ({n i }, {x i }), with n i = n i↑ + n i↓ being the total electron density at site i, and the conjugated fields are (v, η) ≡ ({v i }, {η i }).…”

Supervised learning of few dirty bosons with variable particle number

“…Supervised machine learning is emerging as a potentially disruptive technique to accurately predict the properties of complex quantum systems. It has already allowed researchers to drastically speedup various important computational tasks in quantum chemistry and in condensedmatter physics [1,2], including: molecular dynamics simulations [3][4][5][6][7], electronic structure calculations [8][9][10][11][12][13], structure-based molecular design [14][15][16], and protein-molecule bindingaffinity predictions [17][18][19]. Deep neural networks represent the most powerful and versatile models.…”