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.

中文翻译：

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

相场模型，特别是 Allen-Cahn 型和 Cahn-Hilliard 型方程，已被广泛用于研究界面动力学问题。数十年来，设计用于求解相场模型的精确、高效和稳定的数值算法一直是一个活跃的领域。在本文中，我们通过提出改进的物理信息神经网络 (PINN)，专注于使用深度神经网络为 Allen-Cahn 和 Cahn-Hilliard 方程设计自动数值求解器。尽管 PINN 已被用于研究许多微分方程问题，但我们发现 PINN 在求解相场方程中的直接应用在许多情况下不会提供准确的解。因此，我们提出了各种增加 PINN 逼近能力的技术。作为本文的主要贡献，我们建议在空间和时间上都采用自适应思想并引入各种采样策略，以便我们能够提高 PINN 在求解相场方程时的效率和准确性。此外，改进的 PINN 对偏微分方程的显式形式没有限制，使其适用于更广泛的偏微分方程问题，并在一般情况下阐明了其他偏微分方程的数值近似。