用能量守恒约束提升神经网络求解相场方程的精度
Energy Dissipation Preserving Physics Informed Neural Network for Allen-Cahn Equations
- 将能量耗散特性作为惩罚项加入损失函数，增强模型物理一致性
- 在1~3维空间中保持离散能量单调下降，模拟出相分离与亚稳态现象
- 适用于含随机初值的复杂相场问题，适合做物理引导的深度学习研究
本文基于物理信息神经网络（PINN）研究了具有常数和退化迁移率、多项式与对数能量泛函、确定性和随机初值以及一维、二维和三维空间中带输运项的Allen-Cahn方程的数值解。为提升PINN的建模能力，将Allen-Cahn方程的能量耗散性质作为损失函数中的惩罚项引入。为改善随机初值的学习效果，采用傅里叶级数展开构建初始条件的连续类比。同时结合传统数值分析中的自适应方法以提高所提PINN的有效性。数值结果表明，离散能量始终呈一致下降趋势，并成功再现了相分离与亚稳态等物理现象。
原文摘要 · Abstract (English)
This paper investigates a numerical solution of Allen-Cahn equation with constant and degenerate mobility, with polynomial and logarithmic energy functionals, with deterministic and random initial functions, and with advective term in one, two, and three spatial dimensions, based on the physics-informed neural network (PINN). To improve the learning capacity of the PINN, we incorporate the energy dissipation property of the Allen-Cahn equation as a penalty term into the loss function of the network. To facilitate the learning process of random initials, we employ a continuous analogue of the initial random condition by utilizing the Fourier series expansion. Adaptive methods from traditional numerical analysis are also integrated to enhance the effectiveness of the proposed PINN. Numerical results indicate a consistent decrease in the discrete energy, while also revealing phenomena such as phase separation and metastability.
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