Benefits and pitfalls of symbolic regression as a tool to learn kinetic-energy density functionals

Orbital-free density functional theory can potentially make electronic structure calculations linear scaling. This requires the knowledge of the non-interacting kinetic energy as a functional of the charge density. Such expression should be quantitatively accurate and must admit accurate functional derivatives, so that a minimization procedure can be designed. In this work, symbolic regression is explored as a machine-learning tool to construct non-interacting kinetic-energy functionals, which are defined through analytical expressions so that their functional derivatives can be formally computed. By tracing the Pareto front of the symbolic regression optimization as a function of the number of electrons, we are able to map the transition from the von Weizsäcker functional, exact for the single-electron case, to the Thomas–Fermi one, exact in the homogeneous electron gas limit. Furthermore, extended searches beyond the Pareto front discover more complex functionals. These yield significantly improved kinetic energies, but exhibit unstable functional derivatives. Our work systematically establishes the capabilities and limitations of symbolic regression for constructing semi-local kinetic-energy functionals, highlighting the challenges in constructing simple functionals with simultaneously accurate energies and functional derivatives.