  We numerically study whether there exist nowhere vanishing harmonic $1$-forms
on the real locus of some carefully constructed examples of Calabi-Yau
manifolds, which would then give rise to potentially new examples of
$G_2$-manifolds and an explicit description of their metrics. We do this in two
steps: first, we use a neural network to compute an approximate Calabi-Yau
metric on each manifold. Second, we use another neural network to compute an
approximately harmonic $1$-form with respect to the approximate metric, and
then inspect the found solution. On two manifolds existence of a nowhere
vanishing harmonic $1$-form can be ruled out using differential geometry. The
real locus of a third manifold is diffeomorphic to $S^1 \times S^2$, and our
numerics suggest that when the Calabi-Yau metric is close to a singular limit,
then it admits a nowhere vanishing harmonic $1$-form. We explain how such an
approximate solution could potentially be used in a numerically verified proof
for the fact that our example manifold must admit a nowhere vanishing harmonic
$1$-form.
