  We present a numerical computation, based on neural network techniques, of
the physical Yukawa couplings in a heterotic string theory compactification on
a smooth Calabi-Yau threefold with non-standard embedding. The model belongs to
a large class of heterotic line bundle models that have previously been
identified and whose low-energy spectrum precisely matches that of the MSSM
plus fields uncharged under the Standard Model group. The relevant quantities
for the calculation, that is, the Ricci-flat Calabi-Yau metric, the Hermitian
Yang-Mills bundle metrics and the harmonic bundle-valued forms, are all
computed by training suitable neural networks. For illustration, we consider a
one-parameter family in complex structure moduli space. The computation at each
point along this locus takes about half a day on a single twelve-core CPU. Our
results for the Yukawa couplings are estimated to be within 10\% of the
expected analytic result. We find that the effect of the matter field
normalisation can be significant and can contribute towards generating
hierarchical couplings. We also demonstrate that a zeroth order, semi-analytic
calculation, based on the Fubini-Study metric and its counterparts for the
bundle metric and the bundle-valued forms, leads to roughly correct results,
about 25\% away from the numerical ones. The method can be applied to other
heterotic line bundle models and generalised to other constructions, including
to F-theory models.
