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In this work we explore the potential of a new data-driven approach to thedesign of exchange-correlation (XC) functionals. The approach, inspired byconvolutional filters in computer vision and surrogate functions fromoptimization, utilizes convolutions of the electron density to form a featurespace to represent local electronic environments and neural networks to map thefeatures to the exchange-correlation energy density. These features are orbitalfree, and provide a systematic route to including information at various lengthscales. This work shows that convolutional descriptors are theoreticallycapable of an exact representation of the electron density, and proposesMaxwell-Cartesian spherical harmonic kernels as a class of rotationallyinvariant descriptors for the construction of machine-learned functionals. Theapproach is demonstrated using data from the B3LYP functional on a number ofsmall-molecules containing C, H, O, and N along with a neural networkregression model. The machine-learned functionals are compared to standardphysical approximations and the accuracy is assessed for the absolute energy ofeach molecular system as well as formation energies. The results indicate thatit is possible to reproduce B3LYP formation energies to within chemicalaccuracy using orbital-free descriptors with a spatial extent of 0.2 A. Thefindings provide empirical insight into the spatial range of electron exchange,and suggest that the combination of convolutional descriptors andmachine-learning regression models is a promising new framework for XCfunctional design, although challenges remain in obtaining training data andgenerating models consistent with pseudopotentials.

Design and Analysis of Machine Learning Exchange-Correlation Functionals via Rotationally Invariant Convolutional Descriptors