  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 computationally 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 a pencil 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
numbers and the endomorphism fields of their transcendental lattices.
