We propose Annealed Langevin Monte Carlo for Flow ODE Sampling (ALMC-ODE), a method for generating samples from unnormalized target distributions, with a particular emphasis on multimodal densities that are challenging for standard Markov chain Monte Carlo methods. ALMC-ODE is based on a probability-flow ordinary differential equation (ODE) derived from stochastic interpolants, which continuously transports a standard Gaussian reference distribution at $t = 0$ to the target distribution $ρ$ at $t = 1$. The key innovation lies in an annealed Langevin Markov chain that evolves through a sequence of intermediate distributions bridging the reference and the target. The resulting importance-weighted particles, reweighted via a Jarzynski-based scheme, yield a low-variance estimator of the velocity field governing the ODE. On the theoretical side, we establish a Jarzynski-type reweighting identity for general time-inhomogeneous transition kernels, characterize the optimal backward kernel that minimizes the variance of the importance weights, and prove an $\mathcal{O}(1/n)$ mean squared error bound for the resulting velocity-field estimator. Numerical experiments on challenging benchmarks, including Gaussian mixture models and a 64-dimensional Allen--Cahn field system, demonstrate that ALMC-ODE significantly outperforms both direct Monte Carlo ODE approaches and Hamiltonian Monte Carlo when applied to highly multimodal target distributions.

翻译：我们提出了一种基于退火朗之万蒙特卡洛的流ODE采样方法(ALMC-ODE)，用于从非归一化目标分布中生成样本，特别针对标准马尔可夫链蒙特卡洛方法难以处理的多模态密度分布。ALMC-ODE基于由随机插值导出的概率流常微分方程(ODE)，该方程将$t=0$时刻的标准高斯参考分布连续输运至$t=1$时刻的目标分布$ρ$。其关键创新在于构建一条退火朗之万马尔可夫链，该链通过一系列连接参考分布与目标分布的中间分布逐步演化。所得到的重要性加权粒子经基于Jarzynski方案的重新加权后，可生成控制ODE的速度场低方差估计量。在理论方面，我们针对一般时间非齐次转移核建立了Jarzynski型重加权恒等式，刻画了使重要性权重方差最小化的最优反向核，并证明了所得到的速度场估计量的均方误差界为$\mathcal{O}(1/n)$。在具有挑战性的基准测试（包括高斯混合模型和64维Allen-Cahn场系统）上的数值实验表明，对于高度多模态的目标分布，ALMC-ODE显著优于直接蒙特卡洛ODE方法和哈密顿蒙特卡洛方法。