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Machine Learning Topological Invariants with Neural Networks
Pengfei Zhang, Huitao Shen, Hui Zhai
TL;DR
Topological invariants are nonlocal, nonlinear, and intensive, making topological phases challenging for common machine-learning techniques. This paper trains neural networks on local Hamiltonian inputs for one-dimensional chiral-symmetric band insulators. The convolutional network performs well on larger winding numbers absent from training and computes them through a discrete winding-number formula.
Problem
Learning topological phases is challenging because their invariants are nonlocal, nonlinear, and intensive, properties that make commonly used machine-learning techniques ineffective.
Source: p. 1Method
The study trains neural networks on normalized Hamiltonian inputs sampled at discretized momentum points for one-dimensional chiral-symmetric band insulators.
Source: p. 1Results
The convolutional network performs well on larger, unseen winding numbers and computes them through the discrete winding-number formula.
Source: pp. 3–4Takeaways & Limitations
The authors conclude that their network can be used to analyze data from quantum simulators.
Source: p. 4When training data are restricted to a particular model, the network may learn model-specific features rather than general rules and fail on Hamiltonians lacking inversion symmetry.
Source: p. 2
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.
I. NEURAL NETWORK DETAILS
The network was simplified from a standard multilayer convolutional architecture while performance remained good. Its real-valued output is interpreted as the nearest integer winding number, and test outputs cluster near integers.
Network Design
- The authors reduced a standard multilayer convolutional network’s layers and neurons while performance remained good, stopping when it dropped significantly.The resulting network is reported in the main text.Source: §I · p. 7
Network Output Distribution
- Test outputs ˜w were extremely close to integers, with narrow distribution peaks at each integer across the plotted test sets.The winding number is interpreted as the integer nearest to the network’s real-valued output.Source: §I · p. 7, Figure 1 · p. 7
II. TRAINING DETAILS
The training details report high test accuracy with a smaller Hamiltonian cutoff and robustness to changes in training-data distribution, alongside validation checks and a specified optimization setup.
Training Algorithms
- Training minimized mean squared error with Adam [1], batch size 50, Xavier initialization [2] for weights, and zero-initialized biases.There was no L2, dropout, or other regularization unless specified in the main text; training used Mathematica’s neural-network module, with other hyperparameters at default settings.Source: §II · p. 7
Validation
- Validation used 104 held-out Hamiltonians with winding numbers 0, ±1, and ±2, and the convolutional network showed no overfitting and converged in about 100 epochs.The typical validation loss is shown in Fig. 2.Source: §II · pp. 7–8, Figure 2 · p. 8
Training Data Distribution
- Performance almost did not depend on the training-set distribution across the tested allocations of winding numbers 0, ±1, and ±2.The 10^5-Hamiltonian training set in the main text used proportions 37%, 50%, and 13%, respectively; Fig. 2 shows performance for alternative distributions.Source: §II · p. 8, Figure 2 · p. 8
Training Data Cutoff
- With cutoff c=2, test-set accuracies were 99.3%, 100%, 97.0%, and 98.4% for sets (i)–(iv), comparable to Table I.The winding-number distribution was 26%, 50%, and 23% for 0, ±1, and ±2, respectively.Source: §II · p. 8