用机器学习找物理对偶，自动发现2维伊辛模型的温度映射关系。
Machine learning and optimization-based approaches to duality in statistical physics
- 用神经网络参数化对偶映射，通过相关函数差异优化寻找对偶性。
- 成功复现2维伊辛模型的克兰默斯-万纳尔对偶，精确重建温度对应关系。
- 适合对偶理论、统计物理与机器学习交叉研究者阅读。
对偶性——一个物理系统可用两种不同数学描述——是现代理论物理的核心概念。在格点统计力学中建立对偶性需构造对偶哈密顿量并定义从原模型到对偶模型的观测量映射。本文利用简单神经网络参数化这些映射，并引入损失函数惩罚原模型与对偶模型相关函数之间的差异，将对偶发现过程转化为优化问题。数值求解表明，该框架可成功复现二维伊辛模型的经典克兰默斯-万纳尔对偶，并重建已知的温度映射关系。此外，我们提出一种基于拓扑线映射特性的替代方法，将问题简化为对偶哈密顿量耦合项的优化，并探索了二维伊辛对偶的近邻非最近邻扰动。最后讨论了未来在该框架下发现新对偶的可能性。
原文摘要 · Abstract (English)
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.
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