Neural Mean-Field Games: Extending Mean-Field Game Theory with Neural Stochastic Differential Equations

TL;DR

Neural mean-field games fuse mean-field game theory with neural stochastic differential equations to create data-driven, scalable models of strategic interactions in large populations. By adding neural drift and diffusion at the agent level and training against observed dynamics, the framework yields Nash equilibria that reflect real-world data while maintaining the PDE-based structure of MFGs. The approach is demonstrated on toy (meeting times, El Farol) and real-world (COVID-19 spread in Japan) scenarios, showing that neural dynamics can learn intricate strategies and evolving costs from auxiliary observations. This data-driven, numerically exact method offers a flexible alternative to reinforcement learning for solving mean-field games, with strong potential for epidemiology, economics, and beyond.

Abstract

Mean-field game theory relies on approximating games that are intractible to model due to a very large to infinite population of players. While these kinds of games can be solved analytically via the associated system of partial derivatives, this approach is not model-free, can lead to the loss of the existence or uniqueness of solutions, and may suffer from modelling bias. To reduce the dependency between the model and the game, we introduce neural mean-field games: a combination of mean-field game theory and deep learning in the form of neural stochastic differential equations. The resulting model is data-driven, lightweight, and can learn extensive strategic interactions that are hard to capture using mean-field theory alone. In addition, the model is based on automatic differentiation, making it more robust and objective than approaches based on finite differences. We highlight the efficiency and flexibility of our approach by solving two mean-field games that vary in their complexity, observability, and the presence of noise. Lastly, we illustrate the model's robustness by simulating viral dynamics based on real-world data. Here, we demonstrate that the model's ability to learn from real-world data helps to accurately model the evolution of an epidemic outbreak. Using these results, we show that the model is flexible, generalizable, and requires few observations to learn the distribution underlying the data.

Figures (12)

• Figure 1: An overview of modelling MFGs with neural SDEs. The resulting neural mean-field game combines the MFG mechanics with the neural network output to create more informed strategies.
• Figure 2: The distribution of meeting arrival times for the standard version of the meeting arrival times game. The randomly initialized distribution of arrival times quickly converges to a narrow distribution centred at $\tilde{s}$. The value of $\tilde{s}$ is subject to the Brownian noise of the SDE.
• Figure 3: The distribution of meeting arrival times for the neural MFG version of the meeting arrival times game. The distribution of meeting arrival times of agents whose control is steered towards arriving at 15:00 in turn 15.
• Figure 4: The distribution of probabilities of going to the bar for the standard version of the El Farol Bar problem. In the standard version, the distribution of probabilities converges towards the crowding threshold of $c=0.9$ (blue). The density histograms show the normalized distributions at turn 15.
• Figure 5: The distribution of probabilities of going to the bar for the neural MFG version of the El Farol Bar problem. The neural MFG is trained to align with the observed attendance rates, where the mean $\mu_0=0.2$ is indicated in red. The density histograms show the normalized distributions at turn 15.

Theorems & Definitions (2)

• Theorem 1
• Theorem 2