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Long-Range Machine Learning of Electron Density for Twisted Bilayer Moiré Materials

Zekun Lou1, Alan M. Lewis2, and Mariana Rossi1,3,*

  • *Contact author: mariana.rossi@mpsd.mpg.de
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PRX Intelligence 1, 013015 – Published 27 August, 2026

DOI: https://doi.org/10.1103/4575-9cmx

Abstract

Moiré superlattices in two-dimensional materials exhibit rich quantum phenomena, but ab initio modeling of these systems remains computationally prohibitive. Existing machine learning methods for accelerating density-functional theory can target the prediction of different quantities and often rely on the locality assumption. Here, we train a Gaussian process regression model for symmetry-adapted learning of three-dimensional electron densities exclusively on the electron densities of small displaced bilayer structures and then extrapolate electron density prediction to the large supercells required to describe small twist angles between these bilayers. We show the necessity of long-range descriptors to yield reliable band structures and electrostatic properties of large twisted bilayer structures when these are derived from predicted densities. We demonstrate that the choice of descriptor determines the distribution of residual density errors, which in turn affects the downstream electronic properties. We apply our models to twisted bilayer graphene, hexagonal boron nitride, and transition metal dichalcogenides, focusing on the model’s capacity to predict complex phenomena, including flat-band formation, bandwidth narrowing, domain wall electric fields, and spin-orbit coupling effects. Beyond moiré materials, this approach provides a general methodology for electronic structure prediction in large-scale systems with substantial long-range phenomena related to nonlocal geometric information.

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PRX Intelligence 1, 013015– Published 27 August, 2026
  • Received 12 February 2026
  • Accepted 22 July 2026
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