Symmetry-Constrained Multi-Scale Physics-Informed Neural Networks for Graphene Electronic Band Structure Prediction

TL;DR

This paper introduces a symmetry-constrained multi-scale physics-informed neural network that accurately predicts graphene's electronic band structure while rigorously enforcing crystallographic symmetries, achieving high accuracy and physical consistency.

Contribution

The novel SCMS-PINN architecture integrates multi-head ResNet pathways and hierarchical Dirac constraints to improve band structure predictions with symmetry enforcement.

Findings

Achieved 99.99% reduction in training loss.

Predicted Dirac point gaps within 30.3 μeV of theoretical zero.

Maintained exact symmetry through systematic averaging.

Abstract

Accurate prediction of electronic band structures in two-dimensional materials remains a fundamental challenge, with existing methods struggling to balance computational efficiency and physical accuracy. We present the Symmetry-Constrained Multi-Scale Physics-Informed Neural Network (SCMS-PINN) v35, which directly learns graphene band structures while rigorously enforcing crystallographic symmetries through a multi-head architecture. Our approach introduces three specialized ResNet-6 pathways -- K-head for Dirac physics, M-head for saddle points, and General head for smooth interpolation -- operating on 31 physics-informed features extracted from k-points. Progressive Dirac constraint scheduling systematically increases the weight parameter from 5.0 to 25.0, enabling hierarchical learning from global topology to local critical physics. Training on 10,000 k-points over 300 epochs achieves 99.99\% reduction in training loss (34.597 to 0.003) with validation loss of 0.0085. The model predicts Dirac point gaps within 30.3 $\mu$eV of theoretical zero and achieves average errors of 53.9 meV (valence) and 40.5 meV (conduction) across the Brillouin zone. All twelve C$_{6v}$ operations are enforced through systematic averaging, guaranteeing exact symmetry preservation. This framework establishes a foundation for extending physics-informed learning to broader two-dimensional materials for accelerated discovery.