通过采样求解稳态非线性 Fokker–Planck 方程
Solving stationary nonlinear Fokker-Planck equations via sampling
AI总结：
本文利用相互作用粒子系统与非线性 Fokker–Planck 方程的联系，提出从 N 体 Gibbs 分布采样求稳态解，证明 Gibbs 测度在有界核与非低温条件下收敛，并通过数值实验验证。
摘要：
求解稳态非线性 Fokker–Planck 方程在应用中十分重要，相关例子包括 Poisson–Boltzmann 方程和两层神经网络。利用相互作用粒子系统与非线性 Fokker–Planck 方程之间的联系，我们提出通过从 N 体 Gibbs 分布中采样来求解稳态解。这避免了对 N 体系统进行长时间模拟；更重要的是，这种方法可以避免直接模拟粒子系统时对一致混沌传播的要求。我们证明了当相互作用核有界（不必连续）且温度不太低时，Gibbs 测度收敛到稳态解。针对 Poisson–Boltzmann 方程和两层神经网络进行了数值实验，以验证该方法和理论。
英文摘要：
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.
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