改进PINN方法，让稳态对流主导问题求解更准确
Improving hp-Variational Physics-Informed Neural Networks for Steady-State Convection-Dominated Problems
- 在损失函数中加入类SUPG稳定项，网络自适应预测稳定参数
- 提出新架构学习边界条件指示函数参数，显著提升精度
- 适合对流主导问题的高精度数值模拟研究者使用
本文针对稳态对流-扩散-反应问题，研究了hp-变分物理信息神经网络（FastVPINNs框架）的两种改进方法。首先，在损失函数中引入类SUPG稳定项，并设计网络架构以预测空间变化的稳定参数；其次，针对硬约束狄利克雷边界条件下指示函数选择对解精度影响显著的问题，提出一种可学习指示函数参数的网络架构。数值实验表明，这两种方法相比文献中已有方法均显著提升了计算结果的准确性。
原文摘要 · Abstract (English)
This paper proposes and studies two extensions of applying hp-variational physics-informed neural networks, more precisely the FastVPINNs framework, to convection-dominated convection-diffusion-reaction problems. First, a term in the spirit of a SUPG stabilization is included in the loss functional and a network architecture is proposed that predicts spatially varying stabilization parameters. Having observed that the selection of the indicator function in hard-constrained Dirichlet boundary conditions has a big impact on the accuracy of the computed solutions, the second novelty is the proposal of a network architecture that learns good parameters for a class of indicator functions. Numerical studies show that both proposals lead to noticeably more accurate results than approaches that can be found in the literature.
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