Machine learning band gaps from the electron density

A remarkable consequence of the Hohenberg-Kohn theorem of density functional theory is the existence of an injective map between the electronic density and any observable of the many-electron problem in an external potential. In this work, we study the problem of predicting a particular observable, the band gap of semiconductors and band insulators, from the knowledge of the local electronic density. Using state-of-the-art machine learning techniques, we predict the experimental band gaps from computationally inexpensive density functional theory calculations. We propose a modified Behler-Parrinello (BP) architecture that greatly improves the model capacity while maintaining the symmetry properties of the BP architecture. Using this scheme, we obtain band gaps at a level of accuracy comparable to those obtained with state-of-the-art and computationally intensive hybrid functionals, thus significantly reducing the computational cost of the task.

Figure 1

(a) Histogram of the distribution of experimental band gaps used for training and validation, from Ref. [1]. Histogram box width is 0.5 eV. Colors represent the height of each bin, matching the color labels in Fig. 3. (b) Diagram of the neural network architecture used to approximate the density to band gap map in the specific example of ${\mathrm{TiO}}_2$. Red modules are shared among all atomic species. Blue and yellow modules are specifically applied to Ti and O densities, respectively. From left to right: (i) Rotationally invariant atom-centered density descriptors [see Eq. (3)]; (ii) feature extraction for dimensional reduction; (iii) Behler-Parrinello block and concatenated output $\left\{g_{\alpha }^{\beta }\right\}$; (iv) species-specific Deep Sets neural network and concatenation of output of each species $h_{\beta }$; (v) species-combiner Deep Sets network whose output is the band gap.

Figure 2

Radar plots of the test statistics chosen to measure the band gap prediction accuracy. (a) Train set performance comparison between a Behler-Parrinello architecture—with preprocessing feature extraction—and the full architecture including the Deep Sets modules. Input densities are computed from DFT-PBE. (b) Validation set performance in the full architecture including the Deep Sets modules. Left: Band gaps are learned from atomic positions in the unit cell, from densities computed using DFT-LDA, and from densities computed using DFT-PBE. Right: Comparison of the best performing ML scheme with DFT-calculated band gaps, using LDA, PBE, and HSE06 [6] functionals.

Figure 3

Predicted versus experimental band gaps in the following cases: (a)–(c) Band gaps are computed from DFT calculations using LDA, PBE, and HSE06 [6] functionals. (d) ML-predicted experimental band gaps from PBE-computed densities. Only validation set examples are shown. Orange dashed lines are for visual reference, showing what the exact band gap prediction corresponds to. The color scale for each point corresponds to the number of examples in the training set for that experimental band gap, matching the color scheme of the histogram in Fig. 1. The error statistics of the predicted band gaps shown in this figure are those shown in the right panel of Fig. 2.

Figure 4

(a) Radar plots of the test statistics chosen to measure the band gap prediction accuracy. Validation set performance comparison with the data set with removed examples, as indicated by the color labels. (b) Mean absolute percentage error (MAPE) in the validation set as a function of the number of removed examples expressed as a percentage of the total number of examples of the original data set. (c) Mean percentage error (MPE) in the validation set as a function of the number of removed examples expressed as a percentage of the total number of examples of the original data set.

Figure 5

(a) Absolute error statistics grouped by band gap value with group separations $[0,1,2,3,4,8]$ (eV) as indicated by the quartile Q1 and Q3 caps. (b) Same as panel (a) with the horizontal axis showing the number of examples in each band gap group. It shows that a small number of training examples in a certain band gap range leads to poor performance.