arXiv 2007.157062020-07hep-thcs.LGmath.AG

完全相交Calabi-Yau流形的机器学习方法研究

Machine learning for complete intersection Calabi-Yau manifolds: a methodological study

Università di Torino（都灵大学）；INFN Sezione di Torino（意大利国家核物理研究所都灵分部）；Arnold–Regge Center（阿诺德-雷吉中心）

Harold Erbin, Riccardo Finotello

备注：: 59 pages; additional details, references updated
发表信息：: Phys. Rev. D 103, 126014 (2021)
DOI：: 10.1103/PhysRevD.103.126014

摘要：

我们重新考虑使用机器学习（ML）预测完全相交Calabi-Yau（CICY）3倍的Hodge数$h^{1,1}$和$h^{2,1}$的问题，考虑Candelas-Dale-Lutken-Schimmrigk/Green-Hübsch-Lutken和Anderson-Gao-Gray-Lee分别建立的旧数据集和新数据集。在现实世界的应用程序中，实现ML系统很少会减少向算法提供原始数据。相反，典型的工作流程从探索性数据分析（EDA）开始，其目的是更好地理解输入数据并找到最佳表示。其次是验证程序和基线模型的设计。最后，比较和组合几个ML模型，通常涉及拓扑结构比物理学中通常使用的顺序模型更复杂的神经网络。通过遵循这一过程，我们提高了ML计算霍奇数相对于现有文献的准确性。首先，我们获得了97%（分别为）。$h^{1,1}$使用受旧数据集Inception模型启发的神经网络，准确率仅为30%（分别为99%）。70%的数据用于训练。对于新的一个，简单的线性回归导致几乎100%的准确率，30%的数据用于训练。$h^{2,1}$的计算不太成功，因为我们对两个数据集都只能达到50%的准确率，但这仍然优于使用简单神经网络获得的16%（具有高斯核和特征工程的SVM和顺序卷积网络最多达到36%）。这证明了神经网络对于研究弦论中出现的几何性质是有价值的。

英文摘要：

We revisit the question of predicting both Hodge numbers $h^{1,1}$ and $h^{2,1}$ of complete intersection Calabi-Yau (CICY) 3-folds using machine learning (ML), considering both the old and new datasets built respectively by Candelas-Dale-Lutken-Schimmrigk / Green-Hübsch-Lutken and by Anderson-Gao-Gray-Lee. In real world applications, implementing a ML system rarely reduces to feed the brute data to the algorithm. Instead, the typical workflow starts with an exploratory data analysis (EDA) which aims at understanding better the input data and finding an optimal representation. It is followed by the design of a validation procedure and a baseline model. Finally, several ML models are compared and combined, often involving neural networks with a topology more complicated than the sequential models typically used in physics. By following this procedure, we improve the accuracy of ML computations for Hodge numbers with respect to the existing literature. First, we obtain 97% (resp. 99%) accuracy for $h^{1,1}$ using a neural network inspired by the Inception model for the old dataset, using only 30% (resp. 70%) of the data for training. For the new one, a simple linear regression leads to almost 100% accuracy with 30% of the data for training. The computation of $h^{2,1}$ is less successful as we manage to reach only 50% accuracy for both datasets, but this is still better than the 16% obtained with a simple neural network (SVM with Gaussian kernel and feature engineering and sequential convolutional network reach at best 36%). This serves as a proof of concept that neural networks can be valuable to study the properties of geometries appearing in string theory.

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