论文标题
使用深神经网络解决各向异性扩散方程的加权一阶配方
A weighted first-order formulation for solving anisotropic diffusion equations with deep neural networks
论文作者
论文摘要
在本文中，提出了一种新的加权一阶配方，用于用深神经网络求解各向异性扩散方程。对于许多数值方案，各向异性热通量的准确近似对于总体准确性至关重要。在这项工作中，首先将热通量分解为沿扩散张量的两个特征向量的两个组件，因此各向异性热通量近似被转化为两个各向同性成分的近似值。 此外，为了处理跨界面上扩散张量的跳跃，通过将此一阶配方乘以加权函数来获得加权的一阶配方。通过加权函数的衰减属性，加权的一阶配方总是以方向良好的方式定义。最后，使用深层神经网络近似来解决加权的一阶配方。与原始二阶椭圆公式的神经网络近似相比，提出的方法可以显着提高准确性，尤其是对于不连续的各向异性扩散问题。
In this paper, a new weighted first-order formulation is proposed for solving the anisotropic diffusion equations with deep neural networks. For many numerical schemes, the accurate approximation of anisotropic heat flux is crucial for the overall accuracy. In this work, the heat flux is firstly decomposed into two components along the two eigenvectors of the diffusion tensor, thus the anisotropic heat flux approximation is converted into the approximation of two isotropic components. Moreover, to handle the possible jump of the diffusion tensor across the interface, the weighted first-order formulation is obtained by multiplying this first-order formulation by a weighted function. By the decaying property of the weighted function, the weighted first-order formulation is always well-defined in the pointwise way. Finally, the weighted first-order formulation is solved with deep neural network approximation. Compared to the neural network approximation with the original second-order elliptic formulation, the proposed method can significantly improve the accuracy, especially for the discontinuous anisotropic diffusion problems.
