This paper introduces a new method based on Deep Galerkin Methods (DGMs) for solving high-dimensional stochastic Mean Field Games (MFGs). We achieve this by using two neural networks to approximate the unknown solutions of the MFG system and forward-backward conditions. Our method is efficient, even with a small number of iterations, and is capable of handling up to 300 dimensions with a single layer, which makes it faster than other approaches. In contrast, methods based on Generative Adversarial Networks (GANs) cannot solve MFGs with non-separable Hamiltonians. We demonstrate the effectiveness of our approach by applying it to a traffic flow problem, which was previously solved using the Newton iteration method only in the deterministic case. We compare the results of our method to analytical solutions and previous approaches, showing its efficiency. We also prove the convergence of our neural network approximation with a single hidden layer using the universal approximation theorem.

翻译：本文提出了一种基于深度伽辽金方法(DGMs)的新方法，用于求解高维随机平均场博弈(MFGs)问题。通过使用两个神经网络分别逼近MFG系统的未知解及前向-后向条件，我们的方法即便在较少迭代次数下仍保持高效性，且单层网络即可处理高达300维的问题，使其相较于其他方法具有更快的计算速度。相比之下，基于生成对抗网络(GANs)的方法无法求解具有非可分哈密顿量的MFGs。我们通过将其应用于交通流问题来验证方法的有效性——该问题此前仅能在确定性情形下通过牛顿迭代法求解。我们将本方法的结果与解析解及前人方法进行对比，证明了其高效性。同时，我们基于通用逼近定理，证明了单隐层神经网络近似的收敛性。