We extend the concept of self-consistency for the Fokker-Planck equation (FPE) to the more general McKean-Vlasov equation (MVE). While FPE describes the macroscopic behavior of particles under drift and diffusion, MVE accounts for the additional inter-particle interactions, which are often highly singular in physical systems. Two important examples considered in this paper are the MVE with Coulomb interactions and the vorticity formulation of the 2D Navier-Stokes equation. We show that a generalized self-consistency potential controls the KL-divergence between a hypothesis solution to the ground truth, through entropy dissipation. Built on this result, we propose to solve the MVEs by minimizing this potential function, while utilizing the neural networks for function approximation. We validate the empirical performance of our approach by comparing with state-of-the-art NN-based PDE solvers on several example problems.

翻译：我们将Fokker-Planck方程（FPE）的自洽性概念推广至更一般的McKean-Vlasov方程（MVE）。FPE描述了粒子在漂移和扩散作用下的宏观行为，而MVE则进一步考虑了粒子间的相互交互作用，这种作用在物理系统中往往具有高度奇异性。本文重点研究的两类重要实例包括含库仑相互作用的MVE以及二维Navier-Stokes方程的涡量形式。研究表明，通过熵耗散机制，广义自洽势能可以控制假设解与真实解之间的KL散度。基于这一发现，我们提出通过最小化该势能函数来求解MVE，并利用神经网络进行函数逼近。通过多个算例与当前最优的基于神经网络的偏微分方程求解器进行对比，我们验证了该方法在实际应用中的性能表现。