Prolonged contact between a corrosive liquid and metal alloys can cause
progressive dealloying. For such liquid-metal dealloying (LMD) process, phase
field models have been developed. However, the governing equations often
involve coupled non-linear partial differential equations (PDE), which are
challenging to solve numerically. In particular, stiffness in the PDEs requires
an extremely small time steps (e.g. $10^{-12}$ or smaller). This computational
bottleneck is especially problematic when running LMD simulation until a late
time horizon is required. This motivates the development of surrogate models
capable of leaping forward in time, by skipping several consecutive time steps
at-once. In this paper, we propose U-Shaped Adaptive Fourier Neural Operators
(U-AFNO), a machine learning (ML) model inspired by recent advances in neural
operator learning. U-AFNO employs U-Nets for extracting and reconstructing
local features within the physical fields, and passes the latent space through
a vision transformer (ViT) implemented in the Fourier space (AFNO). We use
U-AFNOs to learn the dynamics mapping the field at a current time step into a
later time step. We also identify global quantities of interest (QoI)
describing the corrosion process (e.g. the deformation of the liquid-metal
interface) and show that our proposed U-AFNO model is able to accurately
predict the field dynamics, in-spite of the chaotic nature of LMD. Our model
reproduces the key micro-structure statistics and QoIs with a level of accuracy
on-par with the high-fidelity numerical solver. We also investigate the
opportunity of using hybrid simulations, in which we alternate forward leap in
time using the U-AFNO with high-fidelity time stepping. We demonstrate that
while advantageous for some surrogate model design choices, our proposed U-AFNO
model in fully auto-regressive settings consistently outperforms hybrid
schemes.

我们提出了 U-AFNO，这是一种受神经运算器学习近期进展启发的机器学习模型，用于学习当前时间步长到后续时间步长的场的动态映射，通过提取和重构物理场中的局部特征，并通过傅立叶空间（AFNO）中的视觉变换器（ViT）传递潜在空间。我们使用 U-AFNO 模型准确预测液态金属腐蚀过程的场动力学，并展示了与高保真数值求解器相当的准确性，同时我们研究了采用 U-AFNO 与高保真时间步恢复相结合的混合模拟的机会，并且我们的 U-AFNO 模型在完全自回归设置中始终优于混合方案。

通过具有 U-Net 主干的混合自适应傅立叶神经运算器加速相场模拟

Accelerating Phase Field Simulations Through a Hybrid Adaptive Fourier  Neural Operator with U-Net Backbone

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。

使用自适应物理信息神经网络求解 Allen-Cahn 和 Cahn-Hilliard 方程

Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive  Physics Informed Neural Networks

In this paper, phase field models are developed for multi-component vesicle
membranes with different lipid compositions and membranes with free boundary.
These models are used to simulate the deformation of membranes under the
elastic bending energy and the line tension energy with prescribed volume and
surface area constraints. By comparing our numerical simulations with recent
experiments, it is demonstrated that the phase field models can capture the
rich phenomena associated with the membrane transformation, thus it offers
great functionality in the simulation and modeling of multicomponent membranes.

使用相场模型模拟研究了多组分囊泡膜的变形并比较了数值模拟和实验结果，其中模型包含弹性弯曲能和线张力能等作用，可以高效地模拟和建模多组分膜的丰富现象。