A hybrid finite-volume scheme with learned rational reconstructions and exact nonlocal fluxes for the space-fractional Burgers equation
Abstract
On coarse grids, fronts of fractional conservation laws can be narrower than one cell, while the flux through each face depends on the whole domain. For the forced space-fractional Burgers equation, we write the finite-volume defect exactly, with the Riesz flux acting as a nonlocal interface flux. Our proposed closure, NRC, is decoder-only, since the encoder is the exact cell-average projection and a small network only maps cell averages to a sub-cell reconstruction. The decoder adds rational atoms with learned complex poles to a trigonometric histopolant, preserves every cell average, and has closed-form advective and Riesz fluxes. We prove that the scheme conserves mass for every value of the weights and that the learned sub-grid flux has a dipole tail. On 64 cells, NRC reaches a median test error of 0.0308% over three seeds, against 0.431% for WENO5-Z and 0.153% for a black-box flux correction. Ablations show that the exact global operator, the projection and the sub-grid Riesz flux carry most of this gain, and most sub-cell fronts carry a deep learned pole within a quarter cell. NRC keeps its advantage over WENO5-Z under shifts of the order and the forcing and, when trained on several diffusivities, under the tested diffusivity shifts by factors from 0.6 to 2.5 in the median over seeds. From 32 to 128 cells, NRC stays an order of magnitude more accurate than WENO5-Z on the same grid. This gain lies in accuracy per degree of freedom and in an interpretable closure.
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BibTeX
@misc{song2026hybrid,
title = {{A hybrid finite-volume scheme with learned rational reconstructions and exact nonlocal fluxes for the space-fractional Burgers equation}},
author = {Zeyuan Song and Zheyu Jiang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31419},
eprint = {2610.31419},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31419},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A hybrid finite-volume scheme with learned rational reconstructions and exact nonlocal fluxes for the space-fractional Burgers equation
AU - Zeyuan Song
AU - Zheyu Jiang
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31419
UR - https://arxiv.org/abs/2610.31419
AB - On coarse grids, fronts of fractional conservation laws can be narrower than one cell, while the flux through each face depends on the whole domain. For the forced space-fractional Burgers equation, we write the finite-volume defect exactly, with the Riesz flux acting as a nonlocal interface flux. Our proposed closure, NRC, is decoder-only, since the encoder is the exact cell-average projection and a small network only maps cell averages to a sub-cell reconstruction. The decoder adds rational atoms with learned complex poles to a trigonometric histopolant, preserves every cell average, and has closed-form advective and Riesz fluxes. We prove that the scheme conserves mass for every value of the weights and that the learned sub-grid flux has a dipole tail. On 64 cells, NRC reaches a median test error of 0.0308% over three seeds, against 0.431% for WENO5-Z and 0.153% for a black-box flux correction. Ablations show that the exact global operator, the projection and the sub-grid Riesz flux carry most of this gain, and most sub-cell fronts carry a deep learned pole within a quarter cell. NRC keeps its advantage over WENO5-Z under shifts of the order and the forcing and, when trained on several diffusivities, under the tested diffusivity shifts by factors from 0.6 to 2.5 in the median over seeds. From 32 to 128 cells, NRC stays an order of magnitude more accurate than WENO5-Z on the same grid. This gain lies in accuracy per degree of freedom and in an interpretable closure.
ER -