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A SAV Neural Network Method for Allen--Cahn and Cahn--Hilliard Equations
Abstract
Phase-field equations are fundamental examples of gradient-flow systems and require numerical solvers that preserve intrinsic energy dissipation. In this work, we propose a neural network-based method that combines an SAV time discretization with a mesh-free neural representation. We establish the unconditional energy stability of the underlying SAV semi-discretization, quantify the energy defect caused by solver residuals, and derive a conditional high-probability error estimate for the Allen--Cahn equation. Numerical experiments show that our approach consistently exhibits energy-stable behavior, while outperforming existing neural network-based solvers.
Related
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BibTeX
@misc{luo2026neural,
title = {{A SAV Neural Network Method for Allen--Cahn and Cahn--Hilliard Equations}},
author = {Tao Luo and Guihong Wang and Wei Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08088},
eprint = {2610.08088},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08088},
}
RIS
TY - UNPB
TI - A SAV Neural Network Method for Allen--Cahn and Cahn--Hilliard Equations
AU - Tao Luo
AU - Guihong Wang
AU - Wei Zhao
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08088
UR - https://arxiv.org/abs/2610.08088
AB - Phase-field equations are fundamental examples of gradient-flow systems and require numerical solvers that preserve intrinsic energy dissipation. In this work, we propose a neural network-based method that combines an SAV time discretization with a mesh-free neural representation. We establish the unconditional energy stability of the underlying SAV semi-discretization, quantify the energy defect caused by solver residuals, and derive a conditional high-probability error estimate for the Allen--Cahn equation. Numerical experiments show that our approach consistently exhibits energy-stable behavior, while outperforming existing neural network-based solvers.
ER -