A PDE-theory-guided neural operator and the long-time relaxation of the two-dimensional anisotropic Boussinesq system
Abstract
We design a neural operator for the two-dimensional Boussinesq system with anisotropic dissipation on the periodic box, guided by a rigorous analysis of its long-time behavior. The operator maps a small perturbation of hydrostatic equilibrium, of amplitude ε, to the vorticity at a late time. We prove that the flow relaxes to a stratified shear flow, with the horizontally averaged vorticity staying within O(ε2) of its initial value ω̅0 and the oscillation about it of order ε. Accordingly, the network outputs ω̅0 plus these two parts, each scaled to its size, which grows as ε2 and ε, respectively. Without this guidance the training either does not converge or misses the ε2 growth. With it, one network reproduces, sample by sample on 2×103 test runs over a factor of $23$ in amplitude, the quantity bounded by the theorem to a median error of $2.2$ percent. Being fast and differentiable with respect to the initial data, the network is also used to search the space of initial data, and with every candidate checked by the solver the search finds initial data for which this quantity is $4.4$ times the largest value among the test runs, raising the lower bound for the constant of the theorem by that factor.
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BibTeX
@misc{moustafa2026pdetheoryguided,
title = {{A PDE-theory-guided neural operator and the long-time relaxation of the two-dimensional anisotropic Boussinesq system}},
author = {Akram Moustafa and Jiahong Wu and Xiaoming Zheng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29938},
eprint = {2610.29938},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29938},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A PDE-theory-guided neural operator and the long-time relaxation of the two-dimensional anisotropic Boussinesq system
AU - Akram Moustafa
AU - Jiahong Wu
AU - Xiaoming Zheng
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29938
UR - https://arxiv.org/abs/2610.29938
AB - We design a neural operator for the two-dimensional Boussinesq system with anisotropic dissipation on the periodic box, guided by a rigorous analysis of its long-time behavior. The operator maps a small perturbation of hydrostatic equilibrium, of amplitude ε, to the vorticity at a late time. We prove that the flow relaxes to a stratified shear flow, with the horizontally averaged vorticity staying within O(ε^2) of its initial value ω̅_0 and the oscillation about it of order ε. Accordingly, the network outputs ω̅_0 plus these two parts, each scaled to its size, which grows as ε^2 and ε, respectively. Without this guidance the training either does not converge or misses the ε^2 growth. With it, one network reproduces, sample by sample on 2×10^3 test runs over a factor of $23$ in amplitude, the quantity bounded by the theorem to a median error of $2.2$ percent. Being fast and differentiable with respect to the initial data, the network is also used to search the space of initial data, and with every candidate checked by the solver the search finds initial data for which this quantity is $4.4$ times the largest value among the test runs, raising the lower bound for the constant of the theorem by that factor.
ER -