论文标题
使用自适应物理知情的神经网络求解Allen-Cahn和Cahn-Hilliard方程
Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networks
论文作者
论文摘要
相位场模型，尤其是Allen-Cahn类型和Cahn-Hilliard类型方程，已被广泛用于研究界面动态问题。设计准确，高效且稳定的数值算法来解决相位场模型已有数十年的活跃场。在本文中，我们专注于使用深层神经网络为Allen-Cahn和Cahn-Hilliard方程设计自动数值求解器，提出改进的物理知情神经网络（PINN）。尽管已经接受了PINN来研究许多微分方程问题，但在许多情况下，我们发现PINN在求解相位方程中的直接应用不会提供准确的解决方案。因此，我们提出了各种技术，以增加PINN的近似能力。作为本文的主要贡献，我们建议在时空和时间上接受自适应思想，并引入各种抽样策略，以便我们能够提高PINN在求解相位场方程方面的效率和准确性。此外，改进的PINN对PDE的明确形式没有限制，使其适用于更广泛的PDE问题，并阐明了其他PDE的数值近似值。
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.
