We introduce a novel unit-time ordinary differential equation (ODE) flow called the preconditioned F\"{o}llmer flow, which efficiently transforms a Gaussian measure into a desired target measure at time 1. To discretize the flow, we apply Euler's method, where the velocity field is calculated either analytically or through Monte Carlo approximation using Gaussian samples. Under reasonable conditions, we derive a non-asymptotic error bound in the Wasserstein distance between the sampling distribution and the target distribution. Through numerical experiments on mixture distributions in 1D, 2D, and high-dimensional spaces, we demonstrate that the samples generated by our proposed flow exhibit higher quality compared to those obtained by several existing methods. Furthermore, we propose leveraging the F\"{o}llmer flow as a warmstart strategy for existing Markov Chain Monte Carlo (MCMC) methods, aiming to mitigate mode collapses and enhance their performance. Finally, thanks to the deterministic nature of the F\"{o}llmer flow, we can leverage deep neural networks to fit the trajectory of sample evaluations. This allows us to obtain a generator for one-step sampling as a result.

翻译：我们提出了一种新颖的单位时间常微分方程流，称为预条件Föllmer流，该流能够在时间1处高效地将高斯测度转化为目标测度。为离散化该流，我们采用欧拉方法，其中速度场可通过解析计算或采用高斯样本的蒙特卡洛近似获得。在合理条件下，我们推导了采样分布与目标分布之间Wasserstein距离的非渐近误差界。通过在一维、二维及高维空间中对混合分布进行数值实验，我们证明所提流生成的样本质量优于多种现有方法。此外，我们建议将Föllmer流作为现有马尔可夫链蒙特卡洛方法的热启动策略，以缓解模式坍塌并提升其性能。最后，得益于Föllmer流的确定性特征，我们可利用深度神经网络拟合样本评估轨迹，从而获得一步采样的生成器。