论文标题
ai增强稳定有限元法
AI-augmented stabilized finite element method
论文作者
论文摘要
提出了一种人工智能的流线上风/彼得罗夫 - 盖尔金有限元方案（AISTAB-FEM），以求解单一扰动的部分微分方程。特别是，提出了一个人工神经网络框架来预测稳定参数的最佳值。神经网络是通过最大程度地减少物理信息成本函数来训练的，在该功能中，将方程式网格和物理参数用作输入特征。此外，将预测的稳定参数用Galerkin溶液的梯度进行标准化，以充分处理边界/内部层区域。所提出的方法抑制了稳定的有限元解决方案中的不足和过冲，并胜过现有的基于神经网络的偏微分方程求解器，例如物理信息的神经网络和变异神经网络。
An artificial intelligence-augmented Streamline Upwind/Petrov-Galerkin finite element scheme (AiStab-FEM) is proposed for solving singularly perturbed partial differential equations. In particular, an artificial neural network framework is proposed to predict optimal values for the stabilization parameter. The neural network is trained by minimizing a physics-informed cost function, where the equation's mesh and physical parameters are used as input features. Further, the predicted stabilization parameter is normalized with the gradient of the Galerkin solution to treat the boundary/interior layer region adequately. The proposed approach suppresses the undershoots and overshoots in the stabilized finite element solution and outperforms the existing neural network-based partial differential equation solvers such as Physics-Informed Neural Networks and Variational Neural Networks.
