Operator Learning for Families of Finite-State Mean-Field Games

2026-02-13

Technology Category

Search and Optimization: Learning to SearchGame Theory and Economic Paradigms: Adversarial LearningMultiagent Systems: Multiagent Learning

Application Category

Security and Privacy: Large-scale security measurementsEconomics, Online Markets and Human Computation: Cost models of using LLMs in production systemsUser Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

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.

Problem

• mean-field games
• operator learning
• Kolmogorov-Fokker-Planck equation
• Hamilton-Jacobi-Bellman equation
• parametric generalization

Innovation

• operator learning
• mean-field games
• flow map
• generalization
• parametric PDEs