Deep unfitted Nitsche method for elliptic interface problems

Abstract

In this paper, we propose a deep unfitted Nitsche method for computing elliptic interface problems with high contrasts in high dimensions. To capture discontinuities of the solution caused by interfaces, we reformulate the problem as an energy minimization involving two weakly coupled components. This enables us to train two deep neural networks to represent two components of the solution in high-dimensional. The curse of dimensionality is alleviated by using the Monte-Carlo method to discretize the unfitted Nitsche energy function. We present several numerical examples to show the efficiency and accuracy of the proposed method.

A Shallow Ritz Method for Elliptic Problems with Singular Sources

“…Another deep least squares method [20] is proposed to minimize the first-order system least-squares functional which is rewritten from a second-order elliptic problem. Comparing with the previous work in [18,19,20], the major difference is that our present network is completely shallow consisting of only one hidden layer. The details will be given in next section.…”

“…In [18] a deep Ritz type approach to solve elliptic interface problem with high-contrast discontinuous coefficients is developed where a shallow neural network is used to approximate the boundary conditions and a deep neural network with ReLU (Rectified Linear Unit) activation function is employed. In [19] a deep unfitted Nitsche method to solve elliptic interface problem with high contrasts in high dimensions is developed where two deep neural networks is formulated to represent two components of the solution. Another deep least squares method [20] is proposed to minimize the first-order system least-squares functional which is rewritten from a second-order elliptic problem.…”