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.

翻译：对偶性概念——即同一物理系统可具有两种不同的数学描述——是现代理论物理的核心思想。在晶格统计力学模型中建立对偶性需要构造对偶哈密顿量以及从原始可观测量到对偶观测量的映射。通过使用简单神经网络参数化这些映射，并引入惩罚原始模型与对偶模型关联函数差异的损失函数，我们将对偶性发现过程表述为优化问题。我们数值求解该问题，并证明该框架能够重新发现二维伊辛模型中著名的Kramers-Wannier对偶性，重构已知的温度映射关系。我们还讨论了另一种利用拓扑线映射已知特征将问题简化为优化对偶哈密顿量耦合参数的方法，并探索了二维伊辛对偶性的次近邻形变。最后，我们探讨了在该框架内发现新对偶性的未来研究方向与前景。