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

TL;DR

This paper extends the Weak Adversarial Neural Pushforward Method to solve both stationary and time-dependent McKean-Vlasov mean-field Fokker-Planck equations, enabling efficient high-dimensional distribution approximation with minimal modifications.

Contribution

It introduces a novel extension of WANPM for nonlinear mean-field equations, including a dimension-dependent initialization rule and efficient sampling strategies.

Findings

Accurate recovery of Gaussian distributions in up to 100 dimensions

Training times range from 27 seconds to 10 minutes on a single GPU

Method effectively handles high-dimensional mean-field problems

Abstract

We extend the Weak Adversarial Neural Pushforward Method (WANPM) to the McKean--Vlasov mean-field Fokker--Planck equation, covering both the stationary and time-dependent cases. The key observation is that the mean-field nonlinearity -- an expectation under the solution distribution -- is naturally estimated by Monte Carlo sampling from the pushforward network, requiring no change to the architecture and only minor modifications to the training loop. For the quadratic (granular media) interaction kernel, the interaction term reduces to the batch sample mean, eliminating secondary sampling entirely. We also identify a dimension-dependent frequency initialization rule for the adversarial test functions, necessary to avoid spurious minimizers. Numerical experiments on linear McKean--Vlasov benchmarks in 2, 5, 20, and 100 dimensions confirm accurate recovery of the exact Gaussian stationary and transient distributions, with training times ranging from 27 seconds (2D) to 10 minutes (100D) on a single GPU.