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Volume 20, Issue 5
Numerical Approximations of the Allen-Cahn-Ohta-Kawasaki Equation with Modified Physics-Informed Neural Networks (PINNs)

Jingjing Xu, Jia Zhao & Yanxiang Zhao

Int. J. Numer. Anal. Mod., 20 (2023), pp. 693-708.

Published online: 2023-09

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  • 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.

  • AMS Subject Headings

65L15, 34L16

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-20-693, author = {Xu , JingjingZhao , Jia and Zhao , Yanxiang}, title = {Numerical Approximations of the Allen-Cahn-Ohta-Kawasaki Equation with Modified Physics-Informed Neural Networks (PINNs)}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2023}, volume = {20}, number = {5}, pages = {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.

}, issn = {2617-8710}, doi = {https://doi.org/10.4208/ijnam2023-1030}, url = {http://global-sci.org/intro/article_detail/ijnam/22008.html} }
TY - JOUR T1 - Numerical Approximations of the Allen-Cahn-Ohta-Kawasaki Equation with Modified Physics-Informed Neural Networks (PINNs) AU - Xu , Jingjing AU - Zhao , Jia AU - Zhao , Yanxiang JO - International Journal of Numerical Analysis and Modeling VL - 5 SP - 693 EP - 708 PY - 2023 DA - 2023/09 SN - 20 DO - http://doi.org/10.4208/ijnam2023-1030 UR - https://global-sci.org/intro/article_detail/ijnam/22008.html KW - Physics-informed neural networks, Allen-Cahn-Ohta-Kawasaki equation, phase field models. AB -

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.

Jingjing Xu, Jia Zhao & Yanxiang Zhao. (2023). Numerical Approximations of the Allen-Cahn-Ohta-Kawasaki Equation with Modified Physics-Informed Neural Networks (PINNs). International Journal of Numerical Analysis and Modeling. 20 (5). 693-708. doi:10.4208/ijnam2023-1030
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