机器学习线丛上同调
Machine Learning Line Bundle Cohomology
摘要：
我们研究了复杂曲面上的线丛上同调以及卡-丘三重上的机器学习的不同方法。综述了基于Logistic S形简单全连接网络的标准函数学习方法，讨论了其主要特点和不足。最近已经观察到，线丛上同调可以通过将Picard格划分为某些区域来描述，在每个区域中，上同调维数由多项式公式来描述。基于这种结构，我们建立了一个网络，能够识别区域及其相关的多项式，从而有效地生成正确的上同调公式的猜想。对于复杂的表面，我们还建立了一个网络，学习某些刚性因子出现在最近发现的上同调维数的主公式。
英文摘要：
We investigate different approaches to machine learning of line bundle cohomology on complex surfaces as well as on Calabi-Yau three-folds. Standard function learning based on simple fully connected networks with logistic sigmoids is reviewed and its main features and shortcomings are discussed. It has been observed recently that line bundle cohomology can be described by dividing the Picard lattice into certain regions in each of which the cohomology dimension is described by a polynomial formula. Based on this structure, we set up a network capable of identifying the regions and their associated polynomials, thereby effectively generating a conjecture for the correct cohomology formula. For complex surfaces, we also set up a network which learns certain rigid divisors which appear in a recently discovered master formula for cohomology dimensions.
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