有限状态均场博弈的算子学习
Operator Learning for Families of Finite-State Mean-Field Games
AI总结：
本文提出了一种算子学习框架，用于求解参数化的有限状态均场博弈，通过理论保证和实证验证实现了对初始分布和终端成本的泛化能力。
摘要：
有限状态均场博弈（MFGs）作为大交互粒子系统的极限而出现，由一个耦合的前向-后向微分方程系统所支配，该系统由一个前向的科尔莫戈罗夫-福克-平面克（KFP）方程描述人口分布，以及一个后向的哈密顿-雅可比-贝尔曼（HJB）方程定义价值函数。高效求解MFG系统具有挑战性，每个系统的结构取决于玩家的初始分布和游戏的终端成本。我们提出了一种算子学习框架，能够求解参数化的MFG家族，使在新的初始分布和终端成本下无需重新训练即可实现泛化。我们基于一个适当定义的流映射的新型正则性结果，提供了对我们的方法的近似误差、参数复杂度和泛化性能的理论保证。我们实证表明，我们的框架在两个代表性的MFG实例上实现了精确的近似：一个网络安全示例和一个高维二次模型，该模型常被用作MFG数值方法的基准测试。
英文摘要：
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.
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