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

arXiv: https://arxiv.org/abs/2508.10718 · HTML: https://mlsanity.com/paper/2508.10718/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 50 references that are in the map.
- First posted on arXiv 14 Aug 2025.

## Builds on (most cited first, 10 of 10)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [SchNet: A continuous-filter convolutional neural network for modeling quantum interactions](https://mlsanity.com/paper/1706.08566/) (2017)
- [IRNet: A General Purpose Deep Residual Regression Framework for Materials Discovery](https://mlsanity.com/paper/1907.03222/) (2019)
- [Predicting Elastic Properties of Materials from Electronic Charge Density Using 3D Deep Convolutional Neural Networks](https://mlsanity.com/paper/2003.13425/) (2020)
- [Deep learning extraction of band structure parameters from density of states: a case study on trilayer graphene](https://mlsanity.com/paper/2210.06310/) (2022)

## Abstract (arXiv)

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 μeV of theoretical zero and achieves average errors of 53.9 meV (valence) and 40.5 meV (conduction) across the Brillouin zone. All twelve C6v 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.