Finding Density Functionals with Machine Learning

Machine learning is used to approximate density functionals. For the model problem of the kinetic energy of noninteracting fermions in 1D, mean absolute errors below $1\mathrm{kcal}/\mathrm{mol}$ on test densities similar to the training set are reached with fewer than 100 training densities. A predictor identifies if a test density is within the interpolation region. Via principal component analysis, a projected functional derivative finds highly accurate self-consistent densities. The challenges for application of our method to real electronic structure problems are discussed.

Figure 1

Comparison of a projected (see within) functional derivative of our MLA with the exact curve.

Figure 2

The shaded region shows the extent of variation of $n(x)$ within our data set for $N=1$. Exact (red, solid) and a self-consistent (black, dashed) density for potential of Fig. 3.

Figure 3

Functional derivative of $T^{\mathrm{ML}}$, evaluated on the density of Fig. 2.

Figure 4

The correlation between MLA error and predictive variance for $N=1$, $M=100$. Each point represents a density in the test set (blue) or new data set (red). The vertical line denotes the transition between interpolation and extrapolation.