Efficient Shallow Ritz Method: 1D Semi-Linear Diffusion-Reaction and Allen-Cahn Equations
Abstract
An efficient shallow Ritz method was developed in [8, 9] for one-dimensional linear diffusion and diffusion-reaction problems. In this paper, the method is extended to a nonlinear diffusion-reaction equation with a cubic reaction term: (1) a basic error estimate of the shallow Ritz discretization is established and (2) a reduced block Newton (rBN) method for solving the resulting non-convex optimization problem is developed. Again, the computational cost of the rBN method is 𝒪(n) for one-dimensional problems, where n is the number of neurons. Combining a convex splitting scheme for time discretization with the efficient shallow Ritz method, we develop a time-stepping neural network method for solving the Allen-Cahn equation. The unconditional energy stability of the time discretization and an error estimate in the L2 norm are established. Numerical experiments demonstrate that the neural network approach yields accurate solutions for singularly perturbed nonlinear diffusion-reaction equations and the Allen-Cahn equation. The method effectively resolves interior and boundary layers without oscillations and significantly outperforms the finite element method on fixed uniform meshes.
Related
- A ghost-cell immersed boundary method for the shallow water equations on Cartesian grids: mass conservation and wetting and drying over variable topography
- Reaction-diffusion equations in periodic media: spreading speeds and spreading sets
- Finite difference methods for three kinds of reaction-diffusion equations with free boundaries
- Convolution Quadrature with Adaptive History Compression for Variable-Order Fractional Reaction--Diffusion Equations
- From Individual-Based Models to General Stochastic Reaction Diffusion Equations
BibTeX
@misc{cai2026efficient,
title = {{Efficient Shallow Ritz Method: 1D Semi-Linear Diffusion-Reaction and Allen-Cahn Equations}},
author = {Zhiqiang Cai and César Herrera},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17553},
eprint = {2610.17553},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17553},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Efficient Shallow Ritz Method: 1D Semi-Linear Diffusion-Reaction and Allen-Cahn Equations
AU - Zhiqiang Cai
AU - César Herrera
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.17553
UR - https://arxiv.org/abs/2610.17553
AB - An efficient shallow Ritz method was developed in [8, 9] for one-dimensional linear diffusion and diffusion-reaction problems. In this paper, the method is extended to a nonlinear diffusion-reaction equation with a cubic reaction term: (1) a basic error estimate of the shallow Ritz discretization is established and (2) a reduced block Newton (rBN) method for solving the resulting non-convex optimization problem is developed. Again, the computational cost of the rBN method is 𝒪(n) for one-dimensional problems, where n is the number of neurons. Combining a convex splitting scheme for time discretization with the efficient shallow Ritz method, we develop a time-stepping neural network method for solving the Allen-Cahn equation. The unconditional energy stability of the time discretization and an error estimate in the L^2 norm are established. Numerical experiments demonstrate that the neural network approach yields accurate solutions for singularly perturbed nonlinear diffusion-reaction equations and the Allen-Cahn equation. The method effectively resolves interior and boundary layers without oscillations and significantly outperforms the finite element method on fixed uniform meshes.
ER -