PreprintCC BY 4.0arXiv
Energetic Variational Deep Learning for Solving High-Dimensional Stationary Fokker-Planck Equations
Abstract
This paper develops a deep learning framework for solving high-dimensional stationary Fokker-Planck equations by combining Monte Carlo weak supervision, energy-variational physics constraints, and quasi-Newton refinement. Monte Carlo statistics anchor the scale of the stationary density, while a free-energy variational loss enforces equilibrium physics; the formulation also applies to stationary systems with pointwise orthogonal rotational components and, using a family of test functions, to systems with more general non-gradient drift. Numerical results demonstrate accurate and robust density approximations across a range of high-dimensional stationary Fokker-Planck problems, including rough-potential and noisy-data settings.
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BibTeX
@misc{kong2026energetic,
title = {{Energetic Variational Deep Learning for Solving High-Dimensional Stationary Fokker-Planck Equations}},
author = {Fanze Kong and Zhouyuan Qin and Shuangquan Xie},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03325},
eprint = {2610.03325},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03325},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Energetic Variational Deep Learning for Solving High-Dimensional Stationary Fokker-Planck Equations
AU - Fanze Kong
AU - Zhouyuan Qin
AU - Shuangquan Xie
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03325
UR - https://arxiv.org/abs/2610.03325
AB - This paper develops a deep learning framework for solving high-dimensional stationary Fokker-Planck equations by combining Monte Carlo weak supervision, energy-variational physics constraints, and quasi-Newton refinement. Monte Carlo statistics anchor the scale of the stationary density, while a free-energy variational loss enforces equilibrium physics; the formulation also applies to stationary systems with pointwise orthogonal rotational components and, using a family of test functions, to systems with more general non-gradient drift. Numerical results demonstrate accurate and robust density approximations across a range of high-dimensional stationary Fokker-Planck problems, including rough-potential and noisy-data settings.
ER -