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：我们研究涉及概率测度Wasserstein空间与函数空间之间算子映射的模型的机器学习任务，例如平均场博弈/控制问题中的情形。基于分箱密度和圆柱逼近的两类神经网络被提出用于学习这类所谓的平均场函数，并通过通用逼近定理提供理论支持。我们针对这两个平均场神经网络的训练进行了多项数值实验，展示了它们在不同测试分布下的泛化误差的准确性和效率。最后，我们提出了基于平均场神经网络求解时间相关平均场问题的不同算法，并通过概率测度Wasserstein空间中半线性偏微分方程示例的数值测试展示了我们的结果。