The Gauss–Manin connection of a family of hypersurfaces governs the change of the period matrix along the family. This connection can be complicated even when the equations defining the family look simple. When this is the case, it is expensive to compute the period matrices of varieties in the family via homotopy continuation. We train neural networks that can quickly and reliably guess the complexity of the Gauss–Manin connection of pencils of hypersurfaces. As an application, we compute the periods of \(96\%\) of smooth quartic surfaces in projective 3-space whose defining equation is a sum of five monomials; from the periods of these quartic surfaces, we extract their Picard lattices and the endomorphism fields of their transcendental lattices.

超曲面族的 Gauss-Manin 连接控制着周期矩阵沿族的变化。即使定义族的方程看起来很简单，这种联系也可能很复杂。在这种情况下，通过同伦延拓计算家族中品种的周期矩阵是昂贵的。我们训练的神经网络可以快速可靠地猜测超曲面铅笔的 Gauss-Manin 连接的复杂性。作为应用，我们计算射影 3 空间中光滑四次曲面的周期\(96\%\)，其定义方程是五个单项式的和；从这些四次曲面的周期中，我们提取了它们的 Picard 格和它们的超越格的自同态场。