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

arXiv: https://arxiv.org/abs/2602.13169 · HTML: https://mlsanity.com/paper/2602.13169/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 41 references that are in the map.
- First posted on arXiv 13 Feb 2026.

## Builds on (most cited first, 26 of 26)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [FourCastNet: Accelerating Global High-Resolution Weather Forecasting using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2208.05419/) (2022)
- [The Shattered Gradients Problem: If resnets are the answer, then what is the question?](https://mlsanity.com/paper/1702.08591/) (2017)
- [A Mathematical Guide to Operator Learning](https://mlsanity.com/paper/2312.14688/) (2023)
- [Approximately Solving Mean Field Games via Entropy-Regularized Deep Reinforcement Learning](https://mlsanity.com/paper/2102.01585/) (2021)
- [On the Convergence of Model Free Learning in Mean Field Games](https://mlsanity.com/paper/1907.02633/) (2019)
- [Convergence Analysis of Machine Learning Algorithms for the Numerical Solution of Mean Field Control and Games: II -- The Finite Horizon Case](https://mlsanity.com/paper/1908.01613/) (2019)
- [Generalization in Mean Field Games by Learning Master Policies](https://mlsanity.com/paper/2109.09717/) (2021)
- [The Parametric Complexity of Operator Learning](https://mlsanity.com/paper/2306.15924/) (2023)
- [Deep Learning Methods for Mean Field Control Problems with Delay](https://mlsanity.com/paper/1905.00358/) (2019)
- [Approximation bounds for norm constrained neural networks with applications to regression and GANs](https://mlsanity.com/paper/2201.09418/) (2022)
- [PPDONet: Deep Operator Networks for Fast Prediction of Steady-State Solutions in Disk-Planet Systems](https://mlsanity.com/paper/2305.11111/) (2023)
- [Learning High-Dimensional McKean-Vlasov Forward-Backward Stochastic Differential Equations with General Distribution Dependence](https://mlsanity.com/paper/2204.11924/) (2022)
- [Signatured Deep Fictitious Play for Mean Field Games with Common Noise](https://mlsanity.com/paper/2106.03272/) (2021)
- [Optimal incentives to mitigate epidemics: a Stackelberg mean field game approach](https://mlsanity.com/paper/2011.03105/) (2020)
- [Mean-field-game model for Botnet defense in Cyber-security](https://mlsanity.com/paper/1511.06642/) (2015)
- [Deep Backward and Galerkin Methods for the Finite State Master Equation](https://mlsanity.com/paper/2403.04975/) (2024)
- [Stochastic Semi-Gradient Descent for Learning Mean Field Games with Population-Aware Function Approximation](https://mlsanity.com/paper/2408.08192/) (2024)
- [Population-aware Online Mirror Descent for Mean-Field Games with Common Noise by Deep Reinforcement Learning](https://mlsanity.com/paper/2509.03030/) (2025)
- [Population-size-Aware Policy Optimization for Mean-Field Games](https://mlsanity.com/paper/2302.03364/) (2023)
- [Analysis of a Finite State Many Player Game Using its Master Equation](https://mlsanity.com/paper/1707.02648/) (2017)
- [Exploiting Approximate Symmetry for Efficient Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2408.15173/) (2024)

## Built on by (most cited first, 2 of 2)

- [Neural Parameter Calibration for Finite-State Mean Field Games](https://mlsanity.com/paper/2606.23155/) (2026)
- [Self-supervised In-context Operator Learning for Stochastic Mean-Field Control](https://mlsanity.com/paper/2608.18282/) (2026)

## Abstract (arXiv)

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.