We study the use of feedforward neural networks for solving parametrized systems of polynomial equations over the real numbers. Our model involves a classifier network to predict the number of real solutions for a given parameter and a regressor network for each realizable number of solutions. We highlight the interaction between algebraic geometry and machine learning throughout. In particular, the classifier training is sensitive to the singularities of the real discriminant locus, and the regressor training is sensitive to the presence of nontrivial real monodromy loops. We demonstrate the model on toy problems as well as systems of equations arising in dynamical systems and robotics.  This is based on joint work with Julianne Barnhart, Jon Hauenstein, Ikenna Nometa, Maggie Regan, Matthew Trang, and Charles Wampler.