A Machine Learning Approach to Duality in Statistical Physics

Prateek Gupta, Andrea E. V. Ferrari, Nabil Iqbal
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:21322-21341, 2025.

Abstract

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 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, numerically reconstructing the known mapping of temperatures. We further investigate the 2d Ising model deformed by a plaquette coupling and find families of “approximate duals”. We discuss future directions and prospects for discovering new dualities within this framework.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-gupta25b, title = {A Machine Learning Approach to Duality in Statistical Physics}, author = {Gupta, Prateek and Ferrari, Andrea E. V. and Iqbal, Nabil}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {21322--21341}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/gupta25b/gupta25b.pdf}, url = {https://proceedings.mlr.press/v267/gupta25b.html}, abstract = {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 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, numerically reconstructing the known mapping of temperatures. We further investigate the 2d Ising model deformed by a plaquette coupling and find families of “approximate duals”. We discuss future directions and prospects for discovering new dualities within this framework.} }
Endnote
%0 Conference Paper %T A Machine Learning Approach to Duality in Statistical Physics %A Prateek Gupta %A Andrea E. V. Ferrari %A Nabil Iqbal %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-gupta25b %I PMLR %P 21322--21341 %U https://proceedings.mlr.press/v267/gupta25b.html %V 267 %X 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 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, numerically reconstructing the known mapping of temperatures. We further investigate the 2d Ising model deformed by a plaquette coupling and find families of “approximate duals”. We discuss future directions and prospects for discovering new dualities within this framework.
APA
Gupta, P., Ferrari, A.E.V. & Iqbal, N.. (2025). A Machine Learning Approach to Duality in Statistical Physics. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:21322-21341 Available from https://proceedings.mlr.press/v267/gupta25b.html.

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