  We construct neural network regression models to predict key metrics of
complexity for Gr\"obner bases of binomial ideals. This work illustrates why
predictions with neural networks from Gr\"obner computations are not a
straightforward process. Using two probabilistic models for random binomial
ideals, we generate and make available a large data set that is able to capture
sufficient variability in Gr\"obner complexity. We use this data to train
neural networks and predict the cardinality of a reduced Gr\"obner basis and
the maximum total degree of its elements. While the cardinality prediction
problem is unlike classical problems tackled by machine learning, our
simulations show that neural networks, providing performance statistics such as
$r^2 = 0.401$, outperform naive guess or multiple regression models with $r^2 =
0.180$.
