A Priori Error Estimation of Physics-Informed Neural Networks Solving Allen--Cahn and Cahn--Hilliard Equations
SandGold AI Research（SandGold AI研究院）；University of Macau（澳门大学）；Xiamen University（厦门大学）；Sichuan University（四川大学）；South China Agricultural University（华南农业大学）；Macau University of Science and Technology（澳门科技大学）
Physics-Informed Neural Networks (PINNs) encounter accuracy limitations when solving the Allen--Cahn (AC) and Cahn--Hilliard (CH) partial differential equations (PDEs). To overcome this, we employ a novel loss function, Residuals-weighted Region Activation Evaluation (Residuals-RAE), featuring a { pre-training weight update scheme}. { Unlike conventional self-adaptive PINNs where weights evolve simultaneously with network parameters, Residuals-RAE-PINNs computes weights from current residuals before each training step and holds them constant during gradient updates. We establish weight convergence under standard neural network optimization assumptions, which justifies analyzing the converged network with constant weights.} Based on this theoretical framework, we derive the error estimation for PINNs with Residuals-RAE when solving AC and CH equations. {The analysis is aligned with Monte-Carlo sampling for the discretization of integrals, consistent with the numerical experiments.} Numerical experiments on one- and two-dimensional AC and CH systems confirm our theoretical results. Additionally, our analysis reveals that feedforward neural networks with two hidden layers and the tanh activation function bound the approximation errors of the solution, its temporal derivative, and the nonlinear term, constrained by the training loss and the number of collocation points.
A New Mixed Finite Element Method For The Cahn-Hilliard Equation
LMAM and School of Mathematical Sciences, Peking University（北京大学数学科学学院）；Chongqing Research Institute of Big Data, Peking University（北京大学重庆大数据研究院）；Beijing Institute of Technology（北京理工大学）；College of Science, Beijing Forestry University（北京林业大学理学院）
Review of thermodynamic structures and structure-preserving discretisations of Cahn--Hilliard-type models
Johannes Gutenberg University Mainz（约翰内斯·古腾堡美因茨大学）；Technical University of Darmstadt（达姆施塔特工业大学）；Radon Institute for Computational and Applied Mathematics（拉登计算与应用数学研究所）；University of Regensburg（雷根斯堡大学）
Numerical approximations of the Cahn-Hilliard and Allen-Cahn Equations with general nonlinear potential using the Invariant Energy Quadratization approach
University of South Carolina（南卡罗来纳大学）；Yantai University（烟台大学）
