发表机构
Center for Science Adventure and Collaborative Research Advancement (SACRA) Graduate School of Science, Kyoto University（京都大学科学冒险与协同研究推进中心（SACRA）理学研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对漂移未知的可逆扩散，提出反向Kolmogorov输运（BKT）方法，利用轨迹估计特征对并沿Wasserstein梯度流移动粒子，无需密度或分数估计即可采样不变律，理论与实验均验证其高精度。
AI 中文摘要
我们考虑一个漂移未知的可逆扩散，通过轨迹观测，并构造其不变律的采样器。轨迹包含该不变律的样本，但基于密度或分数估计构建的采样器会继承这些估计的误差，特别是在亚稳态之间的势垒上。轨迹还包含动力学信息，对于可逆扩散，过程律与其不变律之比满足反向Kolmogorov方程，因此生成元的特征对以闭式形式传播该比值。反向Kolmogorov输运（BKT）利用从轨迹估计的特征对评估该比值，并利用初始粒子一次性计算的系数，将粒子沿Kullback-Leibler散度的Wasserstein梯度流移动。粒子之间不相互作用，BKT定义了一个输运映射，且无需估计密度、分数或漂移。对于精确的特征对，当初始律的谱投影为正时，BKT精确地输运该谱投影。对于估计的特征对，我们证明了一个输运恒等式，其中截断仅通过初始律进入，而估计仅通过特征对的残差进入，并给出了2-Wasserstein距离下的局部稳定性界。利用从输运粒子计算的系数，保留模态的采样误差被抵消，中心极限定理确定了渐近方差。在Ornstein-Uhlenbeck过程、多阱势、可分离乘积和丙氨酸二肽上的实验，包括与基于密度估计的粒子方法的比较，与这些结果一致，并且在谱精确的情况下，BKT的误差达到或低于独立样本的误差。
英文摘要
We consider a reversible diffusion with unknown drift, observed through trajectories, and construct samplers of its invariant law. The trajectories contain samples of this law, but a sampler built from them by density or score estimation inherits the errors of these estimates, in particular on the barriers between metastable states. The trajectories also contain the dynamics, and for a reversible diffusion the ratio between the law of the process and its invariant law solves the backward Kolmogorov equation, so that the eigenpairs of the generator propagate it in closed form. Backward Kolmogorov transport (BKT) evaluates this ratio with eigenpairs estimated from the trajectories and coefficients computed once from the initial particles, and moves the particles along the Wasserstein gradient flow of the Kullback-Leibler divergence. The particles do not interact, BKT defines a transport map, and no density, score or drift is estimated. With exact eigenpairs BKT transports the spectral projection of the initial law exactly when this projection is positive. For estimated eigenpairs we prove a transport identity in which truncation enters only through the initial law and estimation only through the residual of the eigenpairs, and a local stability bound in the 2-Wasserstein distance. With coefficients computed from the transported particles, the sampling error of the retained modes cancels, and a central limit theorem identifies the asymptotic variance. Experiments on Ornstein-Uhlenbeck processes, multi-well potentials, separable products and alanine dipeptide, including comparisons with particle methods based on density estimates, agree with these results, and with an accurate spectrum the errors of BKT reach or fall below those of independent samples.
