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

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 µ eV of theoretical zero and achieves average errors of 53.9 meV (valence) and 40.5 meV (conduction) across the Brillouin zone. This framework establishes a foundation for extending physics-informed learning to broader two-dimensional materials for accelerated discovery. Introduction The accurate prediction of electronic band structures in two-dimensional materials represents a fundamental challenge at the intersection of quantum mechanics, materials science, and machine learning, with profound implications for next-generation electronic and optoelectronic devices [1, 2, 3]. Graphene, the archetypal two-dimensional material, exhibits unique electronic properties arising from its honeycomb lattice structure and linear dispersion relation near the Dirac points, making it both a fascinating subject for fundamental research and a promising candidate for technological applications [4, 5]. Corresponding author Email address: wslee@g.puiching.edu.mo Density functional theory (DFT) has long served as the workhorse for electronic structure calculations, providing reliable results for a wide range of materials systems [6, 7]. Traditional tight-binding approaches offer computational efficiency but fail to capture complex many-body effects, strain-induced modifications, and the subtle interplay between electronic and structural degrees of freedom that are essential for accurate device modeling [10, 11]. Semi-empirical methods attempt to bridge this gap but require extensive parameterization and often lack transferability across different material conditions [12, 13].