Real-space machine learning of correlation density functionals.

Machine learning (ML) plays a pivotal role in extending the reach of quantum chemistry methods for simulating both molecules and materials. However, leveraging ML to overcome the limitations of human-designed density functional approximations (DFAs), the primary workhorse for quantum simulations, remains a major challenge due to their severely limited transferability to unseen chemical systems. Here, we demonstrate how transferability is achieved using real-space ML, where energies are learned point by point in space through energy densities. Central to our real-space learning strategy is the derivation and implementation of correlation energy densities from regularized perturbation theory. This enables two key advances toward constructing highly transferable DFAs, grounded in the Møller-Plesset adiabatic connection framework, for correlation energies defined with respect to the Hartree-Fock reference. First, we introduce the Local Energy Loss, whose data efficiency (expanding each system's single energy into thousands of data points) dramatically enhances transferability when combined with a physically informed ML model. Second, we formulate a real-space, machine-learned, and regularized extension of Spin-Component-Scaled second-order Møller-Plesset perturbation theory, yielding transferable DFAs that effectively mitigate the self-interaction errors common to traditional DFAs.