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.