提出新神经网络方法，确保高维平均场博弈解的数学一致性。
A Constraint-Preserving Neural Network Approach for Solving Mean-Field Games Equilibrium
- 用正则化生成流与时序网络结合求解平均场博弈方程
- 通过密度转移函数约束保持体积守恒与时间连续性
- 适合研究高维博弈、强化学习中的分布演化问题
基于神经网络的方法在求解高维平均场博弈（MFG）均衡方面表现出色，但确保数学上一致的密度耦合演化仍是一大挑战。本文提出NF-MKV Net，将过程正则化归一化流（NF）与状态-策略连接的时间序列神经网络结合，用于求解MKV前向后向随机微分方程（FBSDEs）及其对应的MFG均衡固定点形式。该方法首先将MFG均衡重构为包含密度演化的系数的MKV FBSDEs，嵌入概率框架中；随后利用神经网络近似价值函数及其梯度。为保证体积不变性与时间连续性，对每个密度转移函数施加损失约束。
原文摘要 · Abstract (English)
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.
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