  We study the machine learning task for models with operators mapping between
the Wasserstein space of probability measures and a space of functions, like
e.g. in mean-field games/control problems. Two classes of neural networks,
based on bin density and on cylindrical approximation, are proposed to learn
these so-called mean-field functions, and are theoretically supported by
universal approximation theorems. We perform several numerical experiments for
training these two mean-field neural networks, and show their accuracy and
efficiency in the generalization error with various test distributions.
Finally, we present different algorithms relying on mean-field neural networks
for solving time-dependent mean-field problems, and illustrate our results with
numerical tests for the example of a semi-linear partial differential equation
in the Wasserstein space of probability measures.
