A Priori Error Estimation of Physics-Informed Neural Networks Solving Allen--Cahn and Cahn--Hilliard Equations
<|reference_start|>A Priori Error Estimation of Physics-Informed Neural Networks Solving Allen--Cahn and Cahn--Hilliard Equations: 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.<|reference_end|>
@article{zhang2024a,
    title={A Priori Error Estimation of Physics-Informed Neural Networks Solving
  Allen--Cahn and Cahn--Hilliard Equations},
    author={Guangtao Zhang, Jiani Lin, Qijia Zhai, Huiyu Yang, Xujun Chen,
  Xiaoning Zheng, Ieng Tak Leong},
    journal={arXiv preprint arXiv:2402.02667},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.02667},
    primaryClass={math.NA cs.NA math.OC}
}