This paper aims to analyze errors in the implementation of the Physics-Informed Neural Network (PINN) for solving the Allen--Cahn (AC) and Cahn--Hilliard (CH) partial differential equations (PDEs). The accuracy of PINN is still challenged when dealing with strongly non-linear and higher-order time-varying PDEs. To address this issue, we introduce a stable and bounded self-adaptive weighting scheme known as Residuals-RAE, which ensures fair training and effectively captures the solution. By incorporating this new training loss function, we conduct numerical experiments on 1D and 2D AC and CH systems to validate our theoretical findings. Our theoretical analysis demonstrates that feedforward neural networks with two hidden layers and tanh activation function effectively bound the PINN approximation errors for the solution field, temporal derivative, and nonlinear term of the AC and CH equations by the training loss and number of collocation points.

翻译：本文旨在分析采用物理信息神经网络（PINN）求解Allen-Cahn（AC）和Cahn-Hilliard（CH）偏微分方程（PDEs）时产生的误差。当处理强非线性及高阶时变偏微分方程时，PINN的精度仍面临挑战。为解决该问题，我们引入一种稳定且有界的自适应加权方案——残差自适应估计（Residuals-RAE），该方法确保训练过程的公平性并有效捕捉解的特征。通过融入这一新的训练损失函数，我们在一维和二维AC与CH系统上开展数值实验，以验证理论推导结果。理论分析表明，采用双隐藏层前馈神经网络与tanh激活函数，能够通过训练损失和配置点数量有效界定PINN近似解的误差，涵盖AC和CH方程的解场、时间导数及非线性项。