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 - <p style="text-align: justify;">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.</p>