We present a pseudo-reversible normalizing flow method for efficiently generating samples of the state of a stochastic differential equation (SDE) with different initial distributions. The primary objective is to construct an accurate and efficient sampler that can be used as a surrogate model for computationally expensive numerical integration of SDE, such as those employed in particle simulation. After training, the normalizing flow model can directly generate samples of the SDE's final state without simulating trajectories. Existing normalizing flows for SDEs depend on the initial distribution, meaning the model needs to be re-trained when the initial distribution changes. The main novelty of our normalizing flow model is that it can learn the conditional distribution of the state, i.e., the distribution of the final state conditional on any initial state, such that the model only needs to be trained once and the trained model can be used to handle various initial distributions. This feature can provide a significant computational saving in studies of how the final state varies with the initial distribution. We provide a rigorous convergence analysis of the pseudo-reversible normalizing flow model to the target probability density function in the Kullback-Leibler divergence metric. Numerical experiments are provided to demonstrate the effectiveness of the proposed normalizing flow model.

翻译：我们提出了一种伪可逆归一化流方法，用于高效生成具有不同初始分布的随机微分方程（SDE）状态的样本。其主要目标是构建一个准确且高效的采样器，可作为计算代价高昂的SDE数值积分（如粒子模拟中的积分）的代理模型。训练完成后，该归一化流模型可直接生成SDE最终状态的样本，而无需模拟轨迹。现有针对SDE的归一化流依赖于初始分布，这意味着当初始分布改变时，模型需要重新训练。我们归一化流模型的主要创新在于，它能学习状态的条件分布（即给定任意初始状态下的最终状态分布），从而模型只需训练一次，即可用于处理各种初始分布。这一特性可在研究最终状态随初始分布变化时显著节省计算成本。我们提供了伪可逆归一化流模型在Kullback-Leibler散度度量下收敛于目标概率密度函数的严格收敛性分析。数值实验验证了所提归一化流模型的有效性。