Two Adjoint Perspectives on Fokker-Planck Optimization: A Microscopic-Macroscopic Correspondence
Abstract
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.
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BibTeX
@misc{hellmuth2026adjoint,
title = {{Two Adjoint Perspectives on Fokker-Planck Optimization: A Microscopic-Macroscopic Correspondence}},
author = {Kathrin Hellmuth and Qin Li and Yunan Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02072},
eprint = {2609.02072},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02072},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Two Adjoint Perspectives on Fokker-Planck Optimization: A Microscopic-Macroscopic Correspondence
AU - Kathrin Hellmuth
AU - Qin Li
AU - Yunan Yang
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02072
UR - https://arxiv.org/abs/2609.02072
AB - 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.
ER -