Polarizable atomic multipoles for learning long-range electrostatics

Long-range electrostatics and polarization remain central obstacles to extending machine learning interatomic potentials (MLIPs) to ionic, polar, and interfacial systems. Here, we introduce a semi-local framework for learning electrostatics from energies and forces using polarizable atomic multipoles. Local equivariant descriptors predict environment-dependent latent monopoles, dipoles, and quadrupoles, while residual non-local charge transfer and polarization are captured by non-self-consistent linear response in induced charges and dipoles. Across four diverse benchmarks and four short-range MLIP architectures, the multipole hierarchy and response terms systematically improve potential energy surface accuracy, with the largest gains in systems where long-range effects are essential. More importantly, the learned latent variables recover physically meaningful electrical responses: accurate Born effective charge tensors, emergent polarizabilities, infrared spectra in close agreement with experiments, and semi-quantitative Raman spectra for bulk water and hybrid MAPbI$_3$ perovskite. This systematically improvable, physically transparent framework enables MLIPs trained on standard energy and force labels to predict polarization-sensitive observables.