Mean-field neural networks: Learning mappings on Wasserstein space

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.