Neural network-based methods have demonstrated effectiveness in solving high-dimensional Mean-Field Games (MFG) equilibria, yet ensuring mathematically consistent density-coupled evolution remains a major challenge. This paper proposes the NF-MKV Net, a neural network approach that integrates process-regularized normalizing flow (NF) with state-policy-connected time-series neural networks to solve MKV FBSDEs and their associated fixed-point formulations of MFG equilibria. The method first reformulates MFG equilibria as MKV FBSDEs, embedding density evolution into equation coefficients within a probabilistic framework. Neural networks are then employed to approximate value functions and their gradients. To enforce volumetric invariance and temporal continuity, NF architectures impose loss constraints on each density transfer function.

翻译：基于神经网络的方法在求解高维平均场博弈均衡方面已展现出有效性，但确保数学上一致的密度耦合演化仍是一个主要挑战。本文提出NF-MKV Net，这是一种将过程正则化标准化流与状态-策略关联的时间序列神经网络相结合的神经网络方法，用于求解MKV FBSDE及其相关的平均场博弈均衡不动点表述。该方法首先将平均场博弈均衡重构为MKV FBSDE，在概率框架内将密度演化嵌入方程系数中。随后采用神经网络逼近价值函数及其梯度。为强制体积不变性与时间连续性，NF架构对每个密度转移函数施加损失约束。