A Constraint-Preserving Neural Network Approach for Solving Mean-Field Games Equilibrium

TL;DR

This paper introduces NF-MKV Net, a neural network framework that ensures mathematically consistent density evolution in solving high-dimensional Mean-Field Games equilibria by combining process-regularized normalizing flows with time-series neural networks.

Contribution

The paper presents a novel neural network approach that integrates NF with time-series models to preserve constraints in density evolution for MFG equilibrium solutions.

Findings

Successfully reformulates MFG as MKV FBSDEs with density embedded in coefficients.

Imposes volumetric invariance and temporal continuity constraints on density transfer functions.

Demonstrates effectiveness in solving high-dimensional MFG problems with consistent density evolution.

Abstract

Neural network-based methods have demonstrated effectiveness in solving high-dimensional Mean-Field Games (MFG) equilibria, yet ensuring mathematically consistent density-coupled evolution remains a major challenge. This paper proposes the NF-MKV Net, a neural network approach that integrates process-regularized normalizing flow (NF) with state-policy-connected time-series neural networks to solve MKV FBSDEs and their associated fixed-point formulations of MFG equilibria. The method first reformulates MFG equilibria as MKV FBSDEs, embedding density evolution into equation coefficients within a probabilistic framework. Neural networks are then employed to approximate value functions and their gradients. To enforce volumetric invariance and temporal continuity, NF architectures impose loss constraints on each density transfer function.