We present a novel machine learning moment method for the closure of the moment transport equations associated with the solution of the Williams-Boltzmann equation for polydisperse, evaporating sprays. The method utilizes neural networks to learn optimal closures approximating the dynamics of the kinetic equation using a supervised learning approach. The neural network closure is compared to reference solutions obtained using a Lagrangian random particle method as well as two other state-of-the-art closure models, based on the maximum entropy assumption. Results on 0D and 1D test cases demonstrate that the closures obtained using the machine learning approach is significantly more accurate than the maximum entropy closures with comparable CPU performance. This suggests that such models can be used to replace expensive Lagrangian techniques with similar accuracy at far less cost.