  In this work we explore the potential of a new data-driven approach to the
design of exchange-correlation (XC) functionals. The approach, inspired by
convolutional filters in computer vision and surrogate functions from
optimization, utilizes convolutions of the electron density to form a feature
space to represent local electronic environments and neural networks to map the
features to the exchange-correlation energy density. These features are orbital
free, and provide a systematic route to including information at various length
scales. This work shows that convolutional descriptors are theoretically
capable of an exact representation of the electron density, and proposes
Maxwell-Cartesian spherical harmonic kernels as a class of rotationally
invariant descriptors for the construction of machine-learned functionals. The
approach is demonstrated using data from the B3LYP functional on a number of
small-molecules containing C, H, O, and N along with a neural network
regression model. The machine-learned functionals are compared to standard
physical approximations and the accuracy is assessed for the absolute energy of
each molecular system as well as formation energies. The results indicate that
it is possible to reproduce B3LYP formation energies to within chemical
accuracy using orbital-free descriptors with a spatial extent of 0.2 A. The
findings provide empirical insight into the spatial range of electron exchange,
and suggest that the combination of convolutional descriptors and
machine-learning regression models is a promising new framework for XC
functional design, although challenges remain in obtaining training data and
generating models consistent with pseudopotentials.
