Reinventing Density Functional Theory with Machine Learning on Integral Features

Kohn-Sham density functional theory remains central to electronic structure modeling, because of its efficiency and the steady gains in accuracy that have been achieved through improved density functionals. Here, we introduce a conceptually new strategy in which a multilayer perceptron learns a nonlinear functional of integrated local descriptors ("integral features"). In contrast to grid-wise neural-network approaches, the integral-features formulation applies the nonlinear model only once per calculation, preserving the computational efficiency of standard density functionals while enabling a substantially richer functional dependence. Using this framework, we develop ML25@MN15, an integral-features functional trained on 185 databases spanning thermochemistry, kinetics, noncovalent interactions, spin-flip energies, and molecular structures. Enabled by its expanded representational flexibility, ML25@MN15 achieves a mean unsigned error of 1.05 kcal/mol over 7232 energetic data and consistently outperforms leading modern functionals across all major categories, including systems containing transition metals. Because the neural network operates on integrated descriptors rather than grid-point values, the computational cost of ML25@MN15 remains comparable to MN15. These results demonstrate that introducing integral features and allowing the machine to determine the density functional's dependence on them yields a high-accuracy density functional without increasing computational cost.