Finding Ricci-flat (Calabi-Yau) metrics is a long standing problem in geometry with deep implications for string theory and phenomenology. A new attack on this problem uses neural networks to engineer approximations to the Calabi-Yau metric within a given K\"ahler class. In this paper we investigate numerical Ricci-flat metrics over smooth and singular K3 surfaces and Calabi-Yau threefolds. Using these Ricci-flat metric approximations for the Cefal\'u family of quartic twofolds and the Dwork family of quintic threefolds, we study characteristic forms on these geometries. We observe that the numerical stability of the numerically computed topological characteristic is heavily influenced by the choice of the neural network model, in particular, we briefly discuss a different neural network model, namely Spectral networks, which correctly approximate the topological characteristic of a Calabi-Yau. Using persistent homology, we show that high curvature regions of the manifolds form clusters near the singular points. For our neural network approximations, we observe a Bogomolov--Yau type inequality $3c_2 \geq c_1^2$ and observe an identity when our geometries have isolated $A_1$ type singularities. We sketch a proof that $\chi(X~\smallsetminus~\mathrm{Sing}\,{X}) + 2~|\mathrm{Sing}\,{X}| = 24$ also holds for our numerical approximations.

翻译：寻找里奇平坦（卡拉比-丘）度量是几何学中一个长期存在的问题，对弦理论和现象学具有深远影响。针对该问题的一种新方法利用神经网络在给定凯勒类中构造卡拉比-丘度量的近似解。本文研究了光滑与奇异K3曲面以及卡拉比-丘三维流形上的数值里奇平坦度量。利用Cefalú族四次二维流形和Dwork族五次三维流形的这些里奇平坦度量近似，我们研究了这些几何上的特征形式。我们观察到，数值计算得到的拓扑特征的数值稳定性受到神经网络模型选择的显著影响；特别地，我们简要讨论了另一种神经网络模型，即谱网络，它能正确近似卡拉比-丘的拓扑特征。利用持续同调，我们证明了流形的高曲率区域在奇点附近形成簇。对于我们的神经网络近似，我们观察到Bogomolov--Yau型不等式$3c_2 \geq c_1^2$，并且在几何具有孤立$A_1$型奇点时观察到该等式成立。我们还勾勒出一个证明，表明$\chi(X~\smallsetminus~\mathrm{Sing}\,{X}) + 2~|\mathrm{Sing}\,{X}| = 24$同样适用于我们的数值近似。