Numerical simulation of time fractional Allen-Cahn equation based on Hermite neural solver

Xin Wang, Xiaoping Wang, Haitao Qi and Huanying Xu

Applied Mathematics and Computation, 2025, vol. 491, issue C

Abstract: In this paper, we introduce a high-precision Hermite neural network solver which employs Hermite interpolation technique to construct high-order explicit approximation schemes for fractional derivatives. By automatically satisfying the initial conditions, the construction process of the objective function is simplified, thereby reducing the complexity of the solution. Our neural networks are trained and fine-tuned to solve one-dimensional (1D) and two-dimensional (2D) time fractional Allen-Cahn equations with limited sampling points, yielding high-precision results. Additionally, we tackle the parameter inversion problem by accurately recovering model parameters from observed data, thereby validating the efficacy of the proposed algorithm. We compare the L2 relative error between the exact solution and the predicted solution to verify the robustness and accuracy of the algorithm. This analysis confirms the reliability of our method in capturing the fundamental dynamics of equations. Furthermore, we extend our analysis to three-dimensional (3D) cases, which is covered in the appendix, and provide a thorough evaluation of the performance of our method. This paper also conducts comprehensive analysis of the network structure. Numerical experiments indicate that the number of layers, the number of neurons in each layer, and the choice of learning rate play a crucial role in the performance of our solver.

Keywords: Hermite interpolation; Fractional calculus; Allen-Cahn equation; Neural network; Parameter estimation (search for similar items in EconPapers)
Date: 2025
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0096300324006957
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:491:y:2025:i:c:s0096300324006957

DOI: 10.1016/j.amc.2024.129234

Access Statistics for this article

Applied Mathematics and Computation is currently edited by Theodore Simos

More articles in Applied Mathematics and Computation from Elsevier
Bibliographic data for series maintained by Catherine Liu ().