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.

翻译：Calabi-Yau四流形可作为复维数为5的加权射影空间中的超曲面构造，该空间由6个权重的权重系统定义。本研究实现神经网络以从权重系统学习Calabi-Yau Hodge数，随后通过梯度显著性和符号回归启发式截断Landau-Ginzburg模型公式，用于计算以此方式构造的任意维Calabi-Yau流形的Hodge数。该逼近始终提供紧致的下界，计算速度显著提升（计算时间最多可降低四个数量级），且对大权重系统给出极为精确的结果。此外，本文构建了满足横截性必要非充分条件的互补性权重系统数据集，包括内点性、自反性和内可整除性等性质的考量，并借助机器学习方法进一步确认，从而完成对该权重系统景观的分类。基于此分类知识与所提出逼近的性质，生成了包含7个权重、总权重和$\leq 200$的新型横截权重系统数据集；由此建立新型Calabi-Yau五流形数据库，并计算其相应拓扑性质。进一步，通过近似Hodge数生成候选Calabi-Yau六流形的等效数据库。