  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.
