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

We employ physics-informed neural networks (PINNs) to infer properties of
biological materials using synthetic data. In particular, we successfully apply
PINNs on inferring the thrombus permeability and visco-elastic modulus from
thrombus deformation data, which can be described by the fourth-order
Cahn-Hilliard and Navier-Stokes Equations. In PINNs, the partial differential
equations are encoded into the loss function, where partial derivatives can be
obtained through automatic differentiation (AD). In addition, to tackling the
challenge of calculating the fourth-order derivative in the Cahn-Hilliard
equation with AD, we introduce an auxiliary network along with the main neural
network to approximate the second-derivative of the energy potential term. Our
model can predict simultaneously unknown parameters and velocity, pressure, and
deformation gradient fields by merely training with partial information among
all data, i.e., phase-field and pressure measurements, and is also highly
flexible in sampling within the spatio-temporal domain for data acquisition. We
validate our model by numerical solutions from the spectral/\textit{hp} element
method (SEM) and demonstrate its robustness by training it with noisy
measurements. Our results show that PINNs can accurately infer the material
properties with noisy synthetic data, and thus they have great potential for
inferring these properties from experimental multi-modality and multi-fidelity
data.

本文采用物理信息神经网络 (PINNs) 和合成数据推断生物材料的特性，特别是我们成功地应用于推断血栓渗透性和粘弹性模量，验证了该模型在噪声合成数据中的鲁棒性。