标题： 机器学习和基于优化的方法在统计物理对偶性中的应用 显示英文标题

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

摘要： 对偶性的概念——即一个给定的物理系统可以有两种不同的数学描述——是现代理论物理中的一个关键思想。 在格点统计力学模型中建立对偶性需要构造一个对偶哈密顿量，并从原始可观测量到对偶可观测量的映射。 通过使用简单的神经网络来参数化这些映射，并引入一个损失函数，该函数惩罚原始模型和对偶模型之间关联函数的差异，我们将对偶发现的过程表述为一个优化问题。 我们数值求解这个问题，并表明我们的框架可以重新发现二维伊辛模型的著名克雷默斯-瓦尼尔对偶性，重建已知的温度映射。 我们还讨论了一种替代方法，该方法利用拓扑线映射的已知特征，将问题简化为优化对偶哈密顿量中的耦合项，并探讨二维伊辛对偶性的次近邻变形。 我们讨论了在这个框架内发现新对偶性的未来方向和前景。 显示英文摘要

摘要： 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.