Source-linked AI summary
Machine Learning Topological Invariants with Neural Networks
Pengfei Zhang, Huitao Shen, Hui Zhai
TL;DR
The paper examines how neural networks can be trained and validated to predict winding numbers from Hamiltonian data. Its reported experiments show near-perfect accuracies across four test sets and robustness to training-data distributions and cutoff choices.
Problem
The paper addresses the challenge of training neural networks to predict winding numbers for topological Hamiltonians.
Method
The authors train fully connected and convolutional networks using Adam, mean squared error, mini-batches, validation sets, and varied training-data distributions.
Results
99.3%, 100%, 97.0%, and 98.4% accuracies are reported for test sets (i)–(iv) when training uses cutoff c = 2.
Takeaways & Limitations
The reported performance almost does not depend on the training-data distribution, and the networks show no sign of overfitting.
Abstract
from arXiv · showhide
In this Letter we supervisedly train neural networks to distinguish different topological phases in the context of topological band insulators. After training with Hamiltonians of one-dimensional insulators with chiral symmetry, the neural network can predict their topological winding numbers with nearly 100% accuracy, even for Hamiltonians with larger winding numbers that are not included in the training data. These results show a remarkable success that the neural network can capture the global and nonlinear topological features of quantum phases from local inputs. By opening up the neural network, we confirm that the network does learn the discrete version of the winding number formula. We also make a couple of remarks regarding the role of the symmetry and the opposite effect of regularization techniques when applying machine learning to physical systems.
Supplemental Material for “Machine Learning Topological Invariants with Neural
The supplemental material is for “Machine Learning Topological Invariants with Neural” and lists Pengfei Zhang, Huitao Shen, and Hui Zhai.
- Pengfei Zhang is listed as the first author.
- Huitao Shen and Hui Zhai are also listed as authors.
I. NEURAL NETWORK DETAILS
The supplemental section describes how the network architecture was simplified and how its real-valued winding-number outputs are distributed on test data.
- Network Design: The authors began with a complicated multilayer convolutional network and reduced its complexity until performance dropped significantly.The resulting architecture is reported in the main text.
- Network Output Distribution: The network output is a real number ˜w, interpreted as the nearest integer winding number.
- Network Output Distribution: The convolutional network produces outputs ˜w that are extremely close to integers across different test data sets.Figure 1 shows narrow peaks at each integer.
II. TRAINING DETAILS
The training details specify optimization and validation procedures, then report convergence and robustness to training-data distributions and cutoff choices.
- Training Algorithms: The networks are trained with Adam to minimize mean squared error between ˜w and w using mini-batches of size 50.Weights use Xavier initialization, biases start at zero, and no regularization is used unless specified.
- Validation: The validation set contains 10^4 Hamiltonians with winding numbers w ∈ {0, ±1, ±2} excluded from the training set.
- Validation: The convolutional network shows no sign of overfitting and converges in about 100 epochs.
- Training Data Distribution: The reported performance almost does not depend on the training-data distribution across winding-number classes.The training set contains 10^5 Hamiltonians, with class proportions parameterized as a, 2a, and 1 − 3a.
- Training Data Distribution: 99.3%, 100%, 97.0%, and 98.4% accuracies are obtained on test sets (i)–(iv) when training uses cutoff c = 2.This result is described as comparable with Table I in the main text.