# Machine Learning Topological Invariants with Neural Networks

arXiv: https://arxiv.org/abs/1708.09401 · HTML: https://mlsanity.com/paper/1708.09401/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Aug 2017.

## Built on by (most cited first, 19 of 19)

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- [Can a CNN trained on the Ising model detect the phase transition of the q-state Potts model?](https://mlsanity.com/paper/2104.03632/) (2021)
- [Identifying topology of leaky photonic lattices with machine learning](https://mlsanity.com/paper/2308.14407/) (2023)
- [Topological, or Non-topological? A Deep Learning Based Prediction](https://mlsanity.com/paper/2310.18907/) (2023)
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- [Revisiting the dynamics of Bose-Einstein condensates in a double well by deep learning with a hybrid network](https://mlsanity.com/paper/2104.14657/) (2021)
- [Learning using granularity statistical invariants for classification](https://mlsanity.com/paper/2403.20122/) (2024)
- [Why is topology hard to learn?](https://mlsanity.com/paper/2509.26261/) (2025)
- [Unsupervised Discovery of Intermediate Phase Order in the Frustrated J1-J2 Heisenberg Model via Prometheus Framework](https://mlsanity.com/paper/2602.21468/) (2026)
- [Adjusted Cup-Product Neural Layer](https://mlsanity.com/paper/2606.13568/) (2026)

## Abstract (arXiv)

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.