  We use neural networks to represent the characteristic function of many-body
Gaussian states in the quantum phase space. By a pullback mechanism, we model
transformations due to unitary operators as linear layers that can be cascaded
to simulate complex multi-particle processes. We use the layered neural
networks for non-classical light propagation in random interferometers, and
compute boson pattern probabilities by automatic differentiation. We also
demonstrate that multi-particle events in Gaussian boson sampling can be
optimized by a proper design and training of the neural network weights. The
results are potentially useful to the creation of new sources and complex
circuits for quantum technologies.
