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.

翻译：从伊藤扩散过程中采样不变分布是随机模拟中的一个重要挑战。传统的随机微分方程数值求解器需要精细的步长和漫长的模拟周期，导致样本既存在偏差又具有相关性。当前基于深度学习的方法通过求解稳态福克-普朗克方程，以深度神经网络的形式确定不变概率密度函数，但这些方法通常不直接解决从计算得到的密度函数中采样的问题。在本工作中，我们提出一种框架，采用弱生成采样器直接生成由稳态福克-普朗克方程导出的变换映射所诱导的独立同分布样本。我们提出的损失函数基于福克-普朗克方程的弱形式，结合归一化流来刻画不变分布并促进从基分布生成样本。我们的随机化测试函数避免了传统弱形式中需要迷你-最大优化的需求。与传统生成模型不同，我们的方法既不需要计算计算密集的雅可比行列式，也不要求变换映射的可逆性。我们框架的一个关键组成部分是自适应选择的高斯核函数形式的测试函数族，其中心点从生成的数据样本中选取。在多个基准示例上的实验结果表明，我们的方法具有高效性，既提供了较低的计算成本，又展现出探索多个亚稳态的卓越能力。