# Machine learning and optimization-based approaches to duality in statistical physics

arXiv: https://arxiv.org/abs/2411.04838 · HTML: https://mlsanity.com/paper/2411.04838/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 44 references that are in the map.
- First posted on arXiv 7 Nov 2024.

## Builds on (most cited first, 8 of 8)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Categorical Reparameterization with Gumbel-Softmax](https://mlsanity.com/paper/1611.01144/) (2016)
- [Generative Moment Matching Networks](https://mlsanity.com/paper/1502.02761/) (2015)
- [Inverse statistical problems: from the inverse Ising problem to data science](https://mlsanity.com/paper/1702.01522/) (2017)
- [Quiver Mutations, Seiberg Duality and Machine Learning](https://mlsanity.com/paper/2006.10783/) (2020)
- [Optimal learning of quantum Hamiltonians from high-temperature Gibbs states](https://mlsanity.com/paper/2108.04842/) (2021)
- [Connecting Dualities and Machine Learning](https://mlsanity.com/paper/2002.05169/) (2020)
- [Machine Learning Toric Duality in Brane Tilings](https://mlsanity.com/paper/2409.15251/) (2024)

## Abstract (arXiv)

The notion of duality -- that a given physical system can have two different mathematical descriptions -- is a key idea in modern theoretical physics. Establishing a duality in lattice statistical mechanics models requires the construction of a dual Hamiltonian and a map from the original to the dual observables. By using simple neural networks to parameterize these maps and introducing a loss function that penalises the difference between correlation functions in original and dual models, we formulate the process of duality discovery as an optimization problem. We numerically solve this problem and show that our framework can rediscover the celebrated Kramers-Wannier duality for the 2d Ising model, reconstructing the known mapping of temperatures. We also discuss an alternative approach which uses known features of the mapping of topological lines to reduce the problem to optimizing the couplings in a dual Hamiltonian, and explore next-to-nearest neighbour deformations of the 2d Ising duality. We discuss future directions and prospects for discovering new dualities within this framework.