标题： 四/五/六维卡拉比-丘流形作为 $\mathbb{P}^n_\textbf{w}$ 超曲面：机器学习、逼近与生成 显示英文标题

标题： Calabi-Yau Four/Five/Six-folds as $\mathbb{P}^n_\textbf{w}$ Hypersurfaces: Machine Learning, Approximation, and Generation

摘要： 四维Calabi-Yau流形可以通过在复维度为5的加权射影空间中作为超曲面来构造，这些空间由6个权重定义的权系统确定。 在这项工作中，神经网络被用于从权系统中学习Calabi-Yau的Hodge数，在此过程中，梯度显著性和符号回归启发了Landau-Ginzburg模型公式对于以这种方式构造的任意维Calabi-Yau流形的Hodge数的截断。 该近似总是提供一个紧致的下界，计算速度被证明快得多（计算时间减少了多达四个数量级），并且对于具有大权重的系统给出了非常准确的结果。 此外，还构建了互补的数据集，其中包含满足横截性必要但不充分条件的权系统，包括内部点、反射性和内部分割性质的考虑。 总体而言，生成了这种权系统景观的分类，并通过机器学习方法进一步确认。 利用这一分类的知识和所提出近似的性质，生成了一个由7个权重组成的横截权系统的新数据集，其权重之和为$\leq 200$；从而创建了一个新的Calabi-Yau五维流形数据库，并计算了它们各自的拓扑性质。 此外，还生成了一个候选Calabi-Yau六维流形的等效数据库，并对其Hodge数进行了近似。 显示英文摘要

摘要： Calabi-Yau four-folds may be constructed as hypersurfaces in weighted projective spaces of complex dimension 5 defined via weight systems of 6 weights. In this work, neural networks were implemented to learn the Calabi-Yau Hodge numbers from the weight systems, where gradient saliency and symbolic regression then inspired a truncation of the Landau-Ginzburg model formula for the Hodge numbers of any dimensional Calabi-Yau constructed in this way. The approximation always provides a tight lower bound, is shown to be dramatically quicker to compute (with compute times reduced by up to four orders of magnitude), and gives remarkably accurate results for systems with large weights. Additionally, complementary datasets of weight systems satisfying the necessary but insufficient conditions for transversality were constructed, including considerations of the interior point, reflexivity, and intradivisibility properties. Overall producing a classification of this weight system landscape, further confirmed with machine learning methods. Using the knowledge of this classification, and the properties of the presented approximation, a novel dataset of transverse weight systems consisting of 7 weights was generated for a sum of weights $\leq 200$; producing a new database of Calabi-Yau five-folds, with their respective topological properties computed. Further to this an equivalent database of candidate Calabi-Yau six-folds was generated with approximated Hodge numbers.