Numerical Approximations of the Allen-Cahn-Ohta-Kawasaki Equation with Modified Physics-Informed Neural Networks (PINNs)
Year: 2023
Author: Jingjing Xu, Jia Zhao, Yanxiang Zhao
International Journal of Numerical Analysis and Modeling, Vol. 20 (2023), Iss. 5 : pp. 693–708
Abstract
The physics-informed neural networks (PINNs) has been widely utilized to numerically approximate PDE problems. While PINNs has achieved good results in producing solutions for many partial differential equations, studies have shown that it does not perform well on phase field models. In this paper, we partially address this issue by introducing a modified physics-informed neural networks. In particular, they are used to numerically approximate Allen-Cahn-Ohta-Kawasaki (ACOK) equation with a volume constraint. Technically, the inverse of Laplacian in the ACOK model presents many challenges to the baseline PINNs. To take the zero-mean condition of the inverse of Laplacian, as well as the volume constraint, into consideration, we also introduce a separate neural network, which takes the second set of sampling points in the approximation process. Numerical results are shown to demonstrate the effectiveness of the modified PINNs. An additional benefit of this research is that the modified PINNs can also be applied to learn more general nonlocal phase-field models, even with an unknown nonlocal kernel.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/ijnam2023-1030
International Journal of Numerical Analysis and Modeling, Vol. 20 (2023), Iss. 5 : pp. 693–708
Published online: 2023-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 16
Keywords: Physics-informed neural networks Allen-Cahn-Ohta-Kawasaki equation phase field models.