  Electronic structure theory calculations offer an understanding of matter at
the quantum level, complementing experimental studies in materials science and
chemistry. One of the most widely used methods, density functional theory
(DFT), maps a set of real interacting electrons to a set of fictitious
non-interacting electrons that share the same probability density. Ensuring
that the density remains the same depends on the exchange-correlation (XC)
energy and, by a derivative, the XC potential. Inversions provide a method to
obtain exact XC potentials from target electronic densities, in hopes of
gaining insights into accuracy-boosting approximations. Neural networks provide
a new avenue to perform inversions by learning the mapping from density to
potential. In this work, we learn this mapping using physics-informed machine
learning (PIML) methods, namely physics informed neural networks (PINNs) and
Fourier neural operators (FNOs). We demonstrate the capabilities of these two
methods on a dataset of one-dimensional atomic and molecular models. The
capabilities of each approach are discussed in conjunction with this
proof-of-concept presentation. The primary finding of our investigation is that
the combination of both approaches has the greatest potential for inverting the
Kohn-Sham equations at scale.
