Finite-state mean-field games (MFGs) arise as limits of large interacting particle systems and are governed by an MFG system, a coupled forward-backward differential equation consisting of a forward Kolmogorov-Fokker-Planck (KFP) equation describing the population distribution and a backward Hamilton-Jacobi-Bellman (HJB) equation defining the value function. Solving MFG systems efficiently is challenging, with the structure of each system depending on an initial distribution of players and the terminal cost of the game. We propose an operator learning framework that solves parametric families of MFGs, enabling generalization without retraining for new initial distributions and terminal costs. We provide theoretical guarantees on the approximation error, parametric complexity, and generalization performance of our method, based on a novel regularity result for an appropriately defined flow map corresponding to an MFG system. We demonstrate empirically that our framework achieves accurate approximation for two representative instances of MFGs: a cybersecurity example and a high-dimensional quadratic model commonly used as a benchmark for numerical methods for MFGs.

翻译：有限状态平均场博弈（MFG）作为大型交互粒子系统的极限而出现，其控制方程为一个MFG系统——一个由描述种群分布的前向Kolmogorov-Fokker-Planck（KFP）方程和定义值函数的后向Hamilton-Jacobi-Bellman（HJB）方程构成的耦合前向-后向微分方程。高效求解MFG系统具有挑战性，因为每个系统的结构取决于玩家的初始分布和博弈的终端成本。我们提出了一种算子学习框架，用于求解参数化MFG族，使其能够针对新的初始分布和终端成本实现泛化而无需重新训练。基于对MFG系统对应适当定义的流映射的一个新颖正则性结果，我们为所提方法的逼近误差、参数复杂度和泛化性能提供了理论保证。我们通过实验证明，该框架在两个代表性MFG实例上实现了精确逼近：一个网络安全示例和一个常被用作MFG数值方法基准的高维二次模型。