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Weak Adversarial Neural Pushforward Method for the McKean-Vlasov / Mean-Field Fokker-Planck Equation

Andrew Qing He, Wei Cai

Abstract

We extend the Weak Adversarial Neural Pushforward Method (WANPM) to the McKean-Vlasov mean-field Fokker-Planck equation. For the quadratic interaction kernel, the mean-field nonlinearity reduces to a batch sample mean, requiring no secondary sampling. We focus on the stationary problem, identifying key training subtleties: gradient flow through the self-consistent mean estimate is essential for uniqueness, and adversarial test function frequencies must be initialized at a sufficiently large scale to avoid spurious minimizers. A numerical benchmark on the 1D linear McKean-Vlasov equation confirms accurate recovery of the exact Gaussian stationary distribution.

Weak Adversarial Neural Pushforward Method for the McKean-Vlasov / Mean-Field Fokker-Planck Equation

Abstract

We extend the Weak Adversarial Neural Pushforward Method (WANPM) to the McKean-Vlasov mean-field Fokker-Planck equation. For the quadratic interaction kernel, the mean-field nonlinearity reduces to a batch sample mean, requiring no secondary sampling. We focus on the stationary problem, identifying key training subtleties: gradient flow through the self-consistent mean estimate is essential for uniqueness, and adversarial test function frequencies must be initialized at a sufficiently large scale to avoid spurious minimizers. A numerical benchmark on the 1D linear McKean-Vlasov equation confirms accurate recovery of the exact Gaussian stationary distribution.
Paper Structure (21 sections, 1 theorem, 22 equations, 2 figures, 1 table, 1 algorithm)

This paper contains 21 sections, 1 theorem, 22 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. The McKean--Vlasov Equation
  3. Derivation from an Interacting Particle System
  4. The Aggregation--Diffusion Form
  5. Quadratic Interaction Kernel and Complexity Reduction
  6. WANPM for the Stationary McKean--Vlasov Equation
  7. The Stationary Problem
  8. Weak Formulation
  9. Neural Parametrization
  10. Pushforward network.
  11. Test functions.
  12. Loss Function and Self-Consistency of the Mean Estimate
  13. Adversarial Training and Frequency Initialization
  14. Training Algorithm
  15. Remark on Remark 2 of the companion notes.
  16. ...and 6 more sections

Key Result

Proposition 1

Let $W(\boldsymbol{x}-\boldsymbol{y}) = \frac{1}{2}\|\boldsymbol{x}-\boldsymbol{y}\|^2$. Then

Figures (2)

  • Figure 1: Learned stationary distribution (histogram) vs. exact $\mathcal{N}(0, 0.25)$ (dashed red). Left panel: density comparison. Right panel: Q-Q plot confirming distributional agreement throughout the quantile range. Parameters: $\theta=1$, $\sigma=1$, base distribution $\mathcal{U}[0,1)$, network architecture $(1,10,10,1)$.
  • Figure 2: Training loss convergence. The loss decreases steadily over five orders of magnitude from an initial value near $10^1$ to a final value of $9.6\times10^{-6}$, confirming stable adversarial training dynamics.

Theorems & Definitions (2)

  • Proposition 1: McCann1997
  • proof