Solving the stationary nonlinear Fokker-Planck equations is important in applications and examples include the Poisson-Boltzmann equation and the two layer neural networks. Making use of the connection between the interacting particle systems and the nonlinear Fokker-Planck equations, we propose to solve the stationary solution by sampling from the $N$-body Gibbs distribution. This avoids simulation of the $N$-body system for long time and more importantly such a method can avoid the requirement of uniform propagation of chaos from direct simulation of the particle systems. We establish the convergence of the Gibbs measure to the stationary solution when the interaction kernel is bounded (not necessarily continuous) and the temperature is not very small. Numerical experiments are performed for the Poisson-Boltzmann equations and the two-layer neural networks to validate the method and the theory.

翻译：求解稳态非线性福克-普朗克方程在应用中具有重要意义，例如泊松-玻尔兹曼方程和双层神经网络。利用相互作用粒子系统与非线性福克-普朗克方程之间的联系，我们提出通过从$N$体吉布斯分布中采样来求解稳态解。该方法避免了长时间模拟$N$体系统，更重要的是，它无需像直接模拟粒子系统那样要求均匀的混沌传播。当相互作用核有界（不必连续）且温度不太低时，我们证明了吉布斯测度收敛于稳态解。通过泊松-玻尔兹曼方程和双层神经网络的数值实验验证了该方法和理论的有效性。