Deep Galerkin Method for Mean Field Control Problem

Paper: arXiv 2212.01719

Authors: Jingruo Sun

Abstract

We consider an optimal control problem where the average welfare of weakly interacting agents is of interest. We examine the mean-field control problem as the fluid approximation of the N-agent control problem with the setup of finite-state space, continuous-time, and finite-horizon. The value function of the mean-field control problem is characterized as the unique viscosity solution of a Hamilton-Jacobi-Bellman equation in the simplex. We apply the DGM to estimate the value function and the evolution of the distribution. We also prove the numerical solution approximated by a neural network converges to the analytical solution.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 2.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical framework for mean-field control problems using Hamilton-Jacobi-Bellman equations and proves neural network convergence via the Deep Galerkin Method. However, it lacks empirical validation, backtesting, or application to real-world financial data, making it primarily a mathematical and computational theory contribution.

Research Flowchart

  flowchart TD
    A["Research Goal: Solve N-Agent Mean Field Control Problem"] --> B["Formulate Fluid Approximation: Continuous-Time, Finite-State Space"]
    B --> C["Characterize Value Function via HJB Equation on Simplex"]
    C --> D["Apply Deep Galerkin Method (DGM) Neural Network"]
    D --> E["Simulate & Estimate Value Function and Distribution Evolution"]
    E --> F["Outcome: Proved NN Numerical Solution Converges to Analytical Viscosity Solution"]