Efficient Sampling for Machine Learning Electron Density and Its Response in Real Space

Electron density is a fundamental quantity that can in principle determine all ground state electronic properties of a given system. Although machine learning (ML) models for electron density based on either an atom-centered basis or a real-space grid have been proposed, the demand for a number of high-order basis functions or grid points is enormous. In this work, we propose an efficient grid-point sampling strategy that combines targeted sampling favoring a large density and a screening of grid points associated with linearly independent atomic features. This new sampling strategy is integrated with a field-induced recursively embedded atom neural network model to develop a real-space grid-based ML model for the electron density and its response to an electric field. This approach is applied to a QM9 molecular data set, a H₂O/Pt(111) interfacial system, an Au(100) electrode, and an Au nanoparticle under an electric field. The number of training points is found to be much smaller than previous models, while yielding comparably accurate predictions for the electron density of the entire grid. The resultant machine-learned electron density model enables us to properly partition partial charge onto each atom and analyze the charge variation upon proton transfer in the H₂O/Pt(111) system. The machine-learning electronic response model allows us to predict charge transfer and the electrostatic potential change induced by an electric field applied to an Au(100) electrode or an Au nanoparticle.