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.

翻译：我们研究了涉及算子在概率测度的Wasserstein空间与函数空间之间映射的模型（例如平均场博弈/控制问题）的机器学习任务。基于区间密度近似和柱形近似两类神经网络被提出用于学习这类所谓平均场函数，其理论基础由通用逼近定理提供。我们通过数值实验训练这两类平均场神经网络，并展示了它们在不同测试分布下泛化误差的准确性和高效性。最后，我们提出了基于平均场神经网络求解时变平均场问题的多种算法，并以概率测度Wasserstein空间上半线性偏微分方程为例，通过数值测试验证了所得结果。