Sampling invariant distributions from an Ito diffusion process presents a
significant challenge in stochastic simulation. Traditional numerical solvers
for stochastic differential equations require both a fine step size and a
lengthy simulation period, resulting in both biased and correlated samples.
Current deep learning-based method solves the stationary Fokker--Planck
equation to determine the invariant probability density function in form of
deep neural networks, but they generally do not directly address the problem of
sampling from the computed density function. In this work, we introduce a
framework that employs a weak generative sampler (WGS) to directly generate
independent and identically distributed (iid) samples induced by a
transformation map derived from the stationary Fokker--Planck equation. Our
proposed loss function is based on the weak form of the Fokker--Planck
equation, integrating normalizing flows to characterize the invariant
distribution and facilitate sample generation from the base distribution. Our
randomized test function circumvents the need for mini-max optimization in the
traditional weak formulation. Distinct from conventional generative models, our
method neither necessitates the computationally intensive calculation of the
Jacobian determinant nor the invertibility of the transformation map. A crucial
component of our framework is the adaptively chosen family of test functions in
the form of Gaussian kernel functions with centres selected from the generated
data samples. Experimental results on several benchmark examples demonstrate
the effectiveness of our method, which offers both low computational costs and
excellent capability in exploring multiple metastable states.

从 Fokker-Planck 方程中派生的转换映射，使用弱生成采样器（WGS）直接生成相互独立和同分布（iid）样本的框架，融合标准化流以表征不变分布并促进基础分布的样本生成。实验结果表明，我们的方法在低计算成本和探索多个亚稳态方面具有较好的能力。