Efficient learning of a one-dimensional density functional theory

Density functional theory underlies the most successful and widely used numerical methods for electronic structure prediction of solids. However, it has the fundamental shortcoming that the universal density functional is unknown. In addition, the computational result—energy and charge density distribution of the ground state—is useful for electronic properties of solids mostly when reduced to a band structure interpretation based on the Kohn-Sham approach. Here, we demonstrate how machine learning algorithms can help to free density functional theory from these limitations. We study a theory of spinless fermions on a one-dimensional lattice. The density functional is implicitly represented by a neural network, which predicts, besides the ground-state energy and density distribution, density-density correlation functions. At no point do we require a band structure interpretation. The training data, obtained via exact diagonalization, feeds into a learning scheme inspired by active learning, which minimizes the computational costs for data generation. We show that the network results are of high quantitative accuracy and, despite learning on random potentials, capture both symmetry-breaking and topological phase transitions correctly.

Figure 1

(a) Schematic of the dense neural network used to learn the map from unit-cell potentials and filling to ground-state energy and density-density correlators. (b) Data generation inspired by active learning allows to check the energy deviations of different system sizes and continuing to larger systems only if necessary. (c) Exact versus predicted ground-state energies on the test data set of the actively learning network (ALN). The inset shows the absolute error on the test-energy values as a density histogram $n$. (d) Mean absolute error per correlator entry on the test data set for the ALN. (e) Exact versus predicted ground-state energies for the network trained on a single system size (PLN), with the inset showing the density $n$ of absolute errors on the test data set. (f) Mean absolute error per correlator entry on the test data set for the PLN.

Figure 2

Neural network results for a transition between different (obstructed) atomic limit insulators. (a) Compressibility $\kappa$ for various potential strengths at quarter filling, as calculated from the actively and passively learned neural network, exact diagonalization (ED) and density matrix renormalization group (DMRG) of several system sizes. (b) Schematic depiction of the potential in the four-site unit cell: depending on the strength and sign, two obstructed atomic limits and a metallic phase can be realized. (c) Corresponding observable $C$ as calculated from the $8\times 8$ density-density correlator for the same numerical methods as used for the compressibility. The insets show the correlator as obtained from the ALN in the first (lower left) and second atomic limit (upper right) with the unit cell depicted in red.

Figure 3

Neural network results for a spontaneous symmetry-breaking phase. (a) Ground-state energy for $V=0$ for several electron fillings $n_{\mathrm{e}}^{}$ as calculated from the actively and passively learned neural network and ED. The inset displays the noninteracting band structure. (b) Ground-state energy as a function of the electron filling $n_{\mathrm{e}}^{}$ in the symmetry broken phase ($V=4$). The kink at $n_{\mathrm{e}}=1$ signals an interaction-induced incompressible phase. The inset reveals the band flattening of the noninteracting system. (c) Phase diagram, schematic of the potential, and density-density correlation functions. The latter are obtained from the ALN for $V=0$ (left) and 4 (right). Off-diagonal terms, whose inequivalence signals the symmetry-breaking phase (right) are highlighted in red and green.

Figure 4

Mean absolute error of neural network predictions for various datasets, trained on examples generated with different threshold $\theta$ as illustrated in Table 4. Random potentials were used to generate the evaluation data, with a $N_{\mathrm{uc}}=5,\left|V_i^{}\right|\le 4$ dataset consisting of 5600 samples, $N_{\mathrm{uc}}=6,\left|V_i^{}\right|\le 4$ containing 220 datapoints and 1205 $N_{\mathrm{uc}}=5,\left|V_i^{}\right|\le 1$ input - output pairs. The evaluation sets were not used in the training process of the neural networks.

Figure 5

(a) Finite size scaling plot of the critical potential $V_c$, obtained with a small potential $\delta V$ to break the ground-state degeneracy. Density matrix renormalization group (DMRG) calculations of open boundary conditions are combined with the exact diagonalization (ED) of a 16 site system. (b) Schematic of the potential landscape in the unit cell. (c) Finite size scaling plot of the critical potential $V_c$, obtained with an additional half unit cell and electron to break the ground-state degeneracy. Density matrix renormalization group (DMRG) calculations of open boundary conditions are combined with the exact diagonalization (ED) of an 18 site system. (d) Schematic of the potential landscape with additional two lattice sites.