[论文解读] Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networks

Colby Wight, Jia Zhao|arXiv (Cornell University)|Jul 9, 2020

Model Reduction and Neural Networks参考文献 28被引用 32

一句话总结

该论文通过引入自适应时空采样和训练策略，提升了用于相场方程（Allen–Cahn 与 Cahn–Hilliard）的物理信息神经网络（PINNs）的准确性和效率。

ABSTRACT

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.

研究动机与目标

• Motivate improved numerical solution of phase-field models (Allen–Cahn and Cahn–Hilliard) using neural networks.
• Identify limitations of baseline PINNs in handling sharp interfaces and time evolution in phase-field problems.
• Develop adaptive sampling and time-adaptive strategies to enhance PINN accuracy and efficiency.
• Demonstrate applicability to higher dimensions and complex geometries through benchmark problems.

提出的方法

• Review baseline PINN formulation and apply it to Burgers’ equation as motivation.
• Introduce weighted loss to emphasize initial-time learning and enforce dissipative dynamics.
• Employ mini-batching to improve convergence of the PINN optimization.
• Develop adaptive collocation point sampling in space to focus on moving interfaces via f-network error indicators.
• Propose two time-adaptive strategies: (i) adaptive time/space sampling within time intervals, (ii) time-marching with separate networks per time step.
• Apply adaptive time/space strategies to Allen–Cahn and Cahn–Hilliard equations in 1D, 2D, and 3D settings, including benchmark drops and L-shaped domains.

实验结果

研究问题

• RQ1 Can adaptive sampling in space and time improve PINN accuracy for Allen–Cahn and Cahn–Hilliard equations?
• RQ2 How do weighted loss and mini-batching affect convergence and accuracy for phase-field PINNs?
• RQ3 Are time-adaptive strategies effective for resolving moving sharp interfaces and high-order derivatives in CH equations?
• RQ4 What is the performance of adaptive PINNs on higher-dimensional and complex geometries for phase-field models?

主要发现

• Baseline PINN struggles to accurately solve the Allen–Cahn equation.
• Adding a loss-weighting term improves initial learning but may not achieve full convergence.
• Mini-batching provides some convergence benefits and accuracy improvements.
• Adaptive collocation point resampling significantly improves accuracy (relative l2 error down to 2.33e-2) while using far fewer collocation points (2,000 vs 10,000).
• Time-adaptive strategies (adaptive time intervals and time-marching) enable accurate solutions for difficult cases (e.g., AC with gamma2=4) where fixed-time methods fail.
• Adaptive time-marching and time-adaptive sampling yield accurate 2D and 3D Allen–Cahn results, including shrinking drop benchmarks and complex L-shaped domains.
• Cahn–Hilliard experiments demonstrate the feasibility of the adaptive PINN framework on higher-order PDEs (not fully detailed in the excerpt).