标题： 机器学习晶格图中的环面对偶性 显示英文标题

标题： Machine Learning Toric Duality in Brane Tilings

摘要： 我们将各种机器学习方法应用于研究4d$\mathcal{N}=1$量子场论中的Seiberg对偶性，这些理论出现在D3-膜探测toric Calabi-Yau 3-fold的体积上。 这些理论可以用称为branes tilings或dimer models的环面二分镶嵌来优雅地描述。 一个复杂的红外对偶网络连接了这些理论的空间，并将其划分为普适类，预测和分类这些类是一个自然适合机器学习研究的问题。 在本文中，我们处理了一组初步的此类问题。 我们首先训练一个全连接神经网络，以识别在$\mathbb{Z}_m\times\mathbb{Z}_n$空间上的Seiberg对偶理论的类别，并实现了$R^2=0.988$。 然后，我们评估了我们的方法在理论空间扰动下的鲁棒性，并根据神经网络的学习性质讨论了这些结果。 最后，我们使用一种更复杂的残差架构来分类$Y^{6,0}$理论的toric相空间，并预测其toric图中的各个gauged linear$\sigma$-model多重性。 尽管任务具有非平凡的性质，但我们取得了非常准确的结果；即，在固定Kasteleyn矩阵代表的情况下，回归器实现了$0.021$的平均绝对误差。 我们还讨论了当放松这些假设时性能是如何受到影响的。 显示英文摘要

摘要： We apply a variety of machine learning methods to the study of Seiberg duality within 4d $\mathcal{N}=1$ quantum field theories arising on the worldvolumes of D3-branes probing toric Calabi-Yau 3-folds. Such theories admit an elegant description in terms of bipartite tessellations of the torus known as brane tilings or dimer models. An intricate network of infrared dualities interconnects the space of such theories and partitions it into universality classes, the prediction and classification of which is a problem that naturally lends itself to a machine learning investigation. In this paper, we address a preliminary set of such enquiries. We begin by training a fully connected neural network to identify classes of Seiberg dual theories realised on $\mathbb{Z}_m\times\mathbb{Z}_n$ orbifolds of the conifold and achieve $R^2=0.988$. Then, we evaluate various notions of robustness of our methods against perturbations of the space of theories under investigation, and discuss these results in terms of the nature of the neural network's learning. Finally, we employ a more sophisticated residual architecture to classify the toric phase space of the $Y^{6,0}$ theories, and to predict the individual gauged linear $\sigma$-model multiplicities in toric diagrams thereof. In spite of the non-trivial nature of this task, we achieve remarkably accurate results; namely, upon fixing a choice of Kasteleyn matrix representative, the regressor achieves a mean absolute error of $0.021$. We also discuss how the performance is affected by relaxing these assumptions.