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

Jinwei Liu, Lu Ren, Wang Yao, Xiao Zhang

TL;DR

This work addresses solving high-dimensional mean-field game equilibria by reformulating them as MKV forward-backward stochastic differential equations and solving them with a coupled, constraint-preserving neural framework. NF-MKV Net combines process-regularized normalizing flow to model density evolution with time-series neural networks that capture the time-consistent value function and its gradient, enforcing volumetric invariance and temporal continuity. The method is analyzed for discretization and distribution errors and demonstrated on traffic flow, crowd motion, and obstacle-avoidance tasks, where it achieves high accuracy and outperforms OT- or sampling-based baselines in preserving density evolution. The approach offers a scalable, principled route to solving density-coupled MFG equilibria in high dimensions with practical implications for autonomous systems, crowd management, and networked decision-making.

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.

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

TL;DR

This work addresses solving high-dimensional mean-field game equilibria by reformulating them as MKV forward-backward stochastic differential equations and solving them with a coupled, constraint-preserving neural framework. NF-MKV Net combines process-regularized normalizing flow to model density evolution with time-series neural networks that capture the time-consistent value function and its gradient, enforcing volumetric invariance and temporal continuity. The method is analyzed for discretization and distribution errors and demonstrated on traffic flow, crowd motion, and obstacle-avoidance tasks, where it achieves high accuracy and outperforms OT- or sampling-based baselines in preserving density evolution. The approach offers a scalable, principled route to solving density-coupled MFG equilibria in high dimensions with practical implications for autonomous systems, crowd management, and networked decision-making.

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.

Paper Structure

This paper contains 30 sections, 1 theorem, 33 equations, 7 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Formulations of MFG: HJB-FPK Equations and MKV FBSDEs.
  4. Normalizing Flow
  5. Connections and Differences between MFG and OT
  6. Methodology: NF-MKV Net
  7. Modeling Value Function with MKV FBSDEs
  8. Transformation of the value function equation
  9. Construction of the value function neural network
  10. Modeling density distribution with NF
  11. Refinement of the NF architecture
  12. Construction of the marginal distribution NF
  13. Coupling two processes
  14. Analysis of Computational Error and Complexity
  15. Analysis of Error Estimate
  16. ...and 15 more sections

Key Result

Proposition 1

The flow $(\mathcal{L}(X_t))_{0\leq t\leq T}$ given by the marginal distributions of the forward component of any solution is an equilibria of the MFG problem mfg associated with the stochastic control problem con.

Figures (7)

  • Figure 1: Approach Diagram of the NF-MKV Net
  • Figure 2: NF-MKV Net solutions and numerical error of MFG traffic flow with various initial density distribution and diffusion coefficients
  • Figure 3: 2-dimensional crowd motion dynamics flow
  • Figure 4: 50-dimensional crowd motion dynamics flow
  • Figure 5: 2-dimensional crowd motion dynamics flow with an obstacle.
  • ...and 2 more figures

Theorems & Definitions (4)

  • Definition 1: MFG defined by HJB-FPK lasry2007mean
  • Definition 2: MFG defined by Stochastic Process carmona2018probabilistic
  • Proposition 1: Equivalent carmona2018probabilistic
  • Definition 3: Optimal Transport santambrogio2015optimal