A Structure-Preserving Deep Galerkin Method for Nonlinear Coupled Reaction–Diffusion Systems
- 1Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Moscow 119991, Russia
- 2Key Laboratory of Internet Information Retrieval of Hainan Province, Hainan University, Haikou 570228, China
Abstract
Accurate learning-based solvers for coupled reaction–diffusion systems should preserve mathematical structure rather than merely reduce pointwise residuals. We develop a structure-preserving deep Galerkin method that combines weak residual minimization with positivity, mass-balance, and energy-dissipation constraints. Adaptive quadrature concentrates test points near fronts and structural defects, while staged optimization stabilizes stiff training. Under coercivity, one-sided reaction monotonicity, and entropy compatibility, we establish an energy stability estimate, a weak-residual a posteriori error bound, and a unified structural reliability inequality. Numerical studies on pattern formation, nonlinear diffusion, and heterogeneous media show lower reported aggregate solution errors and structural drift than conventional deep Galerkin, variational, and physics-informed neural solvers; no inferential significance claim is made because seed-resolved records are not available in the retained archive. The results support invariant-aware variational learning as a reliable framework for nonlinear PDE approximation.
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BibTeX
@article{li2026structurepreserving,
title = {{A Structure-Preserving Deep Galerkin Method for Nonlinear Coupled Reaction–Diffusion Systems}},
author = {Xinrui Li and Yuheng Li},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14213890},
url = {https://www.mdpi.com/2227-7390/14/21/3890},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - A Structure-Preserving Deep Galerkin Method for Nonlinear Coupled Reaction–Diffusion Systems
AU - Xinrui Li
AU - Yuheng Li
PY - 2026
DA - 2026/10/25
JO - Mathematics
DO - 10.3390/math14213890
UR - https://www.mdpi.com/2227-7390/14/21/3890
AB - Accurate learning-based solvers for coupled reaction–diffusion systems should preserve mathematical structure rather than merely reduce pointwise residuals. We develop a structure-preserving deep Galerkin method that combines weak residual minimization with positivity, mass-balance, and energy-dissipation constraints. Adaptive quadrature concentrates test points near fronts and structural defects, while staged optimization stabilizes stiff training. Under coercivity, one-sided reaction monotonicity, and entropy compatibility, we establish an energy stability estimate, a weak-residual a posteriori error bound, and a unified structural reliability inequality. Numerical studies on pattern formation, nonlinear diffusion, and heterogeneous media show lower reported aggregate solution errors and structural drift than conventional deep Galerkin, variational, and physics-informed neural solvers; no inferential significance claim is made because seed-resolved records are not available in the retained archive. The results support invariant-aware variational learning as a reliable framework for nonlinear PDE approximation.
SN - 2227-7390
ER -