Research map / Architectures and efficiency / CNN design
Learning Chern Numbers of Multiband Topological Insulators with Gauge Equivariant Neural Networks
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Introduces gauge-equivariant bilinear networks (GEBLNet) with a trace normalization layer to predict discrete Chern numbers of multiband topological insulators, trained only on trivial samples and generalizing to nontrivial ones.
Problem
Learning topological invariants (Chern numbers) of multiband topological insulators is hard because the gauge symmetry U(N) acts independently at every grid site, making the effective symmetry group huge; prior deep models only handled a single filled band (U(1)).
Approach
Use a purely local gauge-equivariant network built from gauge-equivariant bilinear layers (GEBL), gauge-equivariant activations (GEAct), and a novel trace normalization layer (TrNorm), whose gauge invariance guarantees the predicted quantity is a topological invariant; train on diagonal U(1)-valued Wilson loops and trivial-Chern samples with a global plus standard-deviation loss, then rescale outp
Key results
- GEBLNet reaches about 95.9% accuracy on a 5x5 grid with 4 bands and 91.7% with 7 bands, dropping to 52.5% at 8 bands.
- Training only on trivial-Chern samples with a rescaling factor gives about 94.1% accuracy on nontrivial 4-band samples, comparable to 95.9% when training on general data.
- Without TrNorm the naive model learns only up to 3 bands; TrNorm enables learning beyond 4 bands; 4D Chern number prediction reaches MAE around 0.25.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use GEBLNet instead of GEConvNet or TrMLP | 2D Chern-number prediction · 2D grids, non-diagonal data | accuracy and robustness | Helped |
| Use positive-kernel GEConvNet instead of GEBLNet | 2D Chern-number prediction · 5×5 grid, 4 filled bands | accuracy and robustness | Hurt |
| Set GEConvNet kernel size to 0 (ablation) | GEConvNet · 2D Chern-number prediction | accuracy and robustness | Helped |
| Use TrMLP instead of GEBLNet | 2D Chern-number prediction · 2D grids | accuracy and robustness | Hurt |
| Remove TrNorm layers (ablation) | GEBLNet · Chern-number learning, 4 or more bands | accuracy | Hurt |
| Add TrNorm layers after nonlinearities (ablation) | GEBLNet · Chern-number learning, more than 4 bands | accuracy | Helped |
| Train only on trivial samples and rescale outputs at evaluation | GEBLNet · 2D Chern number, 5×5 grid, 4 bands | accuracy: 95.9% -> approximately 94.1% | Hurt |
| Replace bilinear layers with a standard MLP | Determinant predictor · Determinant of real 4×4 matrices | relative error: 0.01 -> 1.02 | Hurt |
| Use more than 4 bands | GEBLNet · 2D Chern number, 5×5 grid | accuracy: 91.7 -> 52.5 | Mixed |
| Train on larger grids rather than the smaller-grid baseline (ablation) | GEBLNet · Chern-number prediction on larger grids | accuracy | Helped |
| Use the 4D higher-order Chern-number task | GEBLNet · 4D second-order Chern number | MAE: around 0.25 | Helped |
Surprising findings
- Polynomial (non-gauge-equivariant) bilinear architectures fail to learn determinants of 4x4 matrices beyond small band sizes, and standard MLPs fail even at 4x4 determinant learning.
- Training with a local GEConvNet using positive kernel sizes performed worse than the degenerate kernel-size-0 local version.
Limitations
- Accuracy drops sharply at 8 bands (52.5%), so high-band learning is not fully solved.
- Trained on trivial samples only, the model recovers Chern numbers only up to a global rescaling factor that must be estimated from nontrivial data.
- Higher-dimensional (4D) discrete Chern numbers are not integers, so evaluation uses MAE instead of accuracy.
Experimental setup
Datasets: synthetic random U(N) link configurations diagonal U(1) Wilson loop dataset
Benchmarks: synthetic 2D Chern number datasets on 5x5 grids 4D Chern number datasets (MAE)
Metrics: accuracy (rounded integer Chern number) MAE relative error
Baselines: GEConvNet TrMLP residual MLP polynomial bilinear architectures
Builds on: Lattice Gauge Equivariant Convolutional Neural Networks (LGE-CNN) gauge equivariant bilinear layers
Closest papers by meaning
- Permutation Equivariant Neural Networks for Symmetric Tensors (ICML 2025 · similarity 0.84)
- Lorentz Local Canonicalization: How to make any Network Lorentz-Equivariant (NeurIPS 2025 · similarity 0.84)
- Are High-Degree Representations Really Unnecessary in Equivariant Graph Neural Networks? (NeurIPS 2024 · similarity 0.84)
- Lie Neurons: Adjoint-Equivariant Neural Networks for Semisimple Lie Algebras (ICML 2024 · similarity 0.84)
- Lie Algebra Canonicalization: Equivariant Neural Operators under arbitrary Lie Groups (ICLR 2025 · similarity 0.83)
- Learning the Electronic Hamiltonian of Large Atomic Structures (ICML 2025 · similarity 0.83)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.