  The one-body reduced density matrix (1-RDM) of a many-body system at zero
temperature gives direct access to many observables, such as the charge
density, kinetic energy and occupation numbers. It would be desirable to
express it as a simple functional of the density or of other local observables,
but to date satisfactory approximations have not yet been found. Deep learning
is the state-of the art approach to perform high dimensional regressions and
classification tasks, and is becoming widely used in the condensed matter
community to develop increasingly accurate density functionals. Autoencoders
are deep learning models that perform efficient dimensionality reduction,
allowing the distillation of data to its fundamental features needed to
represent it. By training autoencoders on a large data-set of 1-RDMs from
exactly solvable real-space model systems, and performing principal component
analysis, the machine learns to what extent the data can be compressed and
hence how it is constrained. We gain insight into these machine learned
constraints and employ them to inform approximations to the 1-RDM as a
functional of the charge density. We exploit known physical properties of the
1-RDM in the simplest possible cases to perform feature engineering, where we
inform the structure of the models from known mathematical relations, allowing
us to integrate existing understanding into the machine learning methods. By
comparing various deep learning approaches we gain insight into what physical
features of the density matrix are most amenable to machine learning, utilising
both known and learned characteristics.
