The $L^2$ gradient flow of the Ginzburg-Landau free energy functional leads
to the Allen Cahn equation that is widely used for modeling phase separation.
Machine learning methods for solving the Allen-Cahn equation in its strong form
suffer from inaccuracies in collocation techniques, errors in computing
higher-order spatial derivatives through automatic differentiation, and the
large system size required by the space-time approach. To overcome these
limitations, we propose a separable neural network-based approximation of the
phase field in a minimizing movement scheme to solve the aforementioned
gradient flow problem. At each time step, the separable neural network is used
to approximate the phase field in space through a low-rank tensor decomposition
thereby accelerating the derivative calculations. The minimizing movement
scheme naturally allows for the use of Gauss quadrature technique to compute
the functional. A `$tanh$' transformation is applied on the neural
network-predicted phase field to strictly bounds the solutions within the
values of the two phases. For this transformation, a theoretical guarantee for
energy stability of the minimizing movement scheme is established. Our results
suggest that bounding the solution through this transformation is the key to
effectively model sharp interfaces through separable neural network. The
proposed method outperforms the state-of-the-art machine learning methods for
phase separation problems and is an order of magnitude faster than the finite
element method.

通过提出可分离的神经网络在最小移动方案中近似相场，在梯度流问题中加速导数计算并通过 `tanh` 变换限制解决方案，有效地模拟尖锐界面，从而胜过现有机器学习方法并比有限元方法快一个数量级。

基于可分离神经网络的梯度流相场建模

Gradient Flow Based Phase-Field Modeling Using Separable Neural Networks

Phase field models, in particular, the Allen-Cahn type and Cahn-Hilliard type
equations, have been widely used to investigate interfacial dynamic problems.
Designing accurate, efficient, and stable numerical algorithms for solving the
phase field models has been an active field for decades. In this paper, we
focus on using the deep neural network to design an automatic numerical solver
for the Allen-Cahn and Cahn-Hilliard equations by proposing an improved physics
informed neural network (PINN). Though the PINN has been embraced to
investigate many differential equation problems, we find a direct application
of the PINN in solving phase-field equations won't provide accurate solutions
in many cases. Thus, we propose various techniques that add to the
approximation power of the PINN. As a major contribution of this paper, we
propose to embrace the adaptive idea in both space and time and introduce
various sampling strategies, such that we are able to improve the efficiency
and accuracy of the PINN on solving phase field equations. In addition, the
improved PINN has no restriction on the explicit form of the PDEs, making it
applicable to a wider class of PDE problems, and shedding light on numerical
approximations of other PDEs in general.

本文使用改进物理知识约束神经网络（PINN）来解决 Allen-Cahn 和 Cahn-Hilliard 方程，并采用自适应的时间和空间采样策略来提高精度和效率，并且改进的 PINN 对于 PDEs 的显式形式没有任何限制，因此适用范围更广。