Weak Adversarial Neural Pushforward Method for the McKean-Vlasov / Mean-Field Fokker-Planck Equation
This addresses numerical solution challenges for mean-field equations in physics and machine learning, but it is incremental as it extends an existing method to a specific case.
The paper tackles solving the stationary McKean-Vlasov mean-field Fokker-Planck equation by extending the Weak Adversarial Neural Pushforward Method, achieving accurate recovery of the exact Gaussian stationary distribution in a 1D linear benchmark.
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.