Variational learning of energy-stable macroscopic flow equations from microscopic laws
Abstract
We present a generalized variational approach for constructing energy-stable macroscopic partial differential equations (PDEs) directly from microscopic physical laws. Rather than fitting the PDE evolution for prescribed variables from macroscopic trajectories, we construct an extendable variational structure by adjoining auxiliary variables to seek autonomous reduced dynamics in an enlarged state space. By defining these variables as micro--macro encoders with explicit microscopic counterparts, we cast the learning problem as the sample-wise joint identification of the reduced state representation and the variational ingredients from time-discrete data, enabling systematic law-level transfer across scales. The resulting PDEs preserve frame indifference, physical constraints, and energy stability by construction, and admit a fluctuation--dissipation-consistent stochastic extension. Numerical results for the non-Newtonian hydrodynamics of polymeric fluids show that the learned models capture the viscoelastic responses induced by nonlinear and heterogeneous molecular interactions beyond empirical closures.
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BibTeX
@misc{ge2026variational,
title = {{Variational learning of energy-stable macroscopic flow equations from microscopic laws}},
author = {Pei Ge and Huan Lei},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01432},
eprint = {2611.01432},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01432},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Variational learning of energy-stable macroscopic flow equations from microscopic laws
AU - Pei Ge
AU - Huan Lei
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01432
UR - https://arxiv.org/abs/2611.01432
AB - We present a generalized variational approach for constructing energy-stable macroscopic partial differential equations (PDEs) directly from microscopic physical laws. Rather than fitting the PDE evolution for prescribed variables from macroscopic trajectories, we construct an extendable variational structure by adjoining auxiliary variables to seek autonomous reduced dynamics in an enlarged state space. By defining these variables as micro--macro encoders with explicit microscopic counterparts, we cast the learning problem as the sample-wise joint identification of the reduced state representation and the variational ingredients from time-discrete data, enabling systematic law-level transfer across scales. The resulting PDEs preserve frame indifference, physical constraints, and energy stability by construction, and admit a fluctuation--dissipation-consistent stochastic extension. Numerical results for the non-Newtonian hydrodynamics of polymeric fluids show that the learned models capture the viscoelastic responses induced by nonlinear and heterogeneous molecular interactions beyond empirical closures.
ER -