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 IP, 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四重流形可构造为复维数为5的加权射影空间中的超曲面，其定义依赖于由6个权重构成的权重系统。本研究采用神经网络从权重系统中学习Calabi-Yau Hodge数，通过梯度显著性和符号回归启发，对任意维数下以此方式构造的Calabi-Yau流形的Landau-Ginzburg模型Hodge数公式进行了截断。该近似方法始终提供严格下界，计算速度显著提升（计算时间最多减少四个数量级），且对权重较大的系统给出了极为精确的结果。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，并考虑了IP性、自反性及内整除性等性质，最终完成了对该权重系统景观的分类，并进一步通过机器学习方法验证。基于此分类认知及所提近似性质，生成了包含7个权重且权重和$\leq 200$的新型横截性权重系统数据集，建立了相应Calabi-Yau五重流形数据库并计算其拓扑性质。进一步地，生成了等价的候选Calabi-Yau六重流形数据库，并采用近似Hodge数进行表征。