# Proximal methods for stationary Mean Field Games with local couplings

arXiv: https://arxiv.org/abs/1608.07701 · HTML: https://mlsanity.com/paper/1608.07701/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Aug 2016.

## Built on by (most cited first, 11 of 11)

- [Scalable Deep Reinforcement Learning Algorithms for Mean Field Games](https://mlsanity.com/paper/2203.11973/) (2022)
- [Scaling up Mean Field Games with Online Mirror Descent](https://mlsanity.com/paper/2103.00623/) (2021)
- [Coupled Multiwavelet Neural Operator Learning for Coupled Partial Differential Equations](https://mlsanity.com/paper/2303.02304/) (2023)
- [Mean Field LQG Social Optimization: A Reinforcement Learning Approach](https://mlsanity.com/paper/2410.15119/) (2024)
- [Exploration noise for learning linear-quadratic mean field games](https://mlsanity.com/paper/2107.00839/) (2021)
- [Decoding Mean Field Games from Population and Environment Observations By Gaussian Processes](https://mlsanity.com/paper/2312.06625/) (2023)
- [Learning Surrogate Potential Mean Field Games via Gaussian Processes: A Data-Driven Approach to Ill-Posed Inverse Problems](https://mlsanity.com/paper/2502.11506/) (2025)
- [Learning to Stop: Deep Learning for Mean Field Optimal Stopping](https://mlsanity.com/paper/2410.08850/) (2024)
- [Networked Communication for Decentralised Agents in Mean-Field Games](https://mlsanity.com/paper/2306.02766/) (2023)
- [Gaussian process policy iteration with additive Schwarz acceleration for forward and inverse HJB and mean field game problems](https://mlsanity.com/paper/2505.00909/) (2025)
- [Decentralized Strategies for Finite Population LQG Social Control: A Reinforcement Learning Approach](https://mlsanity.com/paper/2608.30244/) (2026)

## Abstract (arXiv)

We address the numerical approximation of Mean Field Games with local couplings. For power-like Hamiltonians, we consider both unconstrained and constrained stationary systems with density constraints in order to model hard congestion effects. For finite difference discretizations of the Mean Field Game system, we follow a variational approach. We prove that the aforementioned schemes can be obtained as the optimality system of suitably defined optimization problems. In order to prove the existence of solutions of the scheme with a variational argument, the monotonicity of the coupling term is not used, which allow us to recover general existence results. Next, assuming next that the coupling term is monotone, the variational problem is cast as a convex optimization problem for which we study and compare several proximal type methods. These algorithms have several interesting features, such as global convergence and stability with respect to the viscosity parameter, which can eventually be zero. We assess the performance of the methods via numerical experiments.