  We present an application of invariant polynomials in machine learning. Using
the methods developed in previous work, we obtain two types of generators of
the Lorentz- and permutation-invariant polynomials in particle momenta; minimal
algebra generators and Hironaka decompositions. We discuss and prove some
approximation theorems to make use of these invariant generators in machine
learning algorithms in general and in neural networks specifically. By
implementing these generators in neural networks applied to regression tasks,
we test the improvements in performance under a wide range of hyperparameter
choices and find a reduction of the loss on training data and a significant
reduction of the loss on validation data. For a different approach on
quantifying the performance of these neural networks, we treat the problem from
a Bayesian inference perspective and employ nested sampling techniques to
perform model comparison. Beyond a certain network size, we find that networks
utilising Hironaka decompositions perform the best.
