发表机构
California Institute of Technology; University of Wisconsin-Madison; Cornell University（加州理工学院; 威斯康星大学麦迪逊分校; 康奈尔大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对Fokker-Planck优化，协调了宏观与微观两种伴随视角的对应关系，证明离散化后两者梯度仍为一致近似并建立了显式收敛速率。
AI 中文摘要
Fokker-Planck方程既可以通过概率密度给出宏观欧拉描述，也可以通过随机轨迹给出微观拉格朗日描述。因此，受Fokker-Planck方程约束的优化问题可以从任一视角构建。令人惊讶的是，对应的伴随方程看似存在本质差异：宏观伴随由后向Kolmogorov方程支配，而微观伴随沿随机轨迹逐路径演化。在本注记中，我们通过建立连续体框架下的对应关系，协调了这两种形式。我们进一步证明，尽管离散化后两者的离散梯度不再一致，但均能为连续体梯度提供一致的数值近似。我们为两种离散化策略均建立了显式收敛速率。
英文摘要
The Fokker-Planck equation admits both a macroscopic Eulerian description through probability densities and a microscopic Lagrangian description through stochastic trajectories. Consequently, optimization problems constrained by the Fokker-Planck equation can be formulated from either perspective. Surprisingly, the corresponding adjoint equations appear to be fundamentally different: the macroscopic adjoint is governed by the backward Kolmogorov equation, whereas the microscopic adjoint evolves pathwise along stochastic trajectories. In this note, we reconcile these two formulations by establishing their correspondence in the continuum setting. We further show that, although their discrete gradients no longer coincide after discretization, both provide consistent numerical approximations of the continuum gradient. Explicit convergence rates are established for both discretization strategies.
