Deep Unfitted Nitsche Method for Elliptic Interface Problems

Abstract

This paper proposes a deep unfitted Nitsche method for solving 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 problem involving two weakly coupled components. This enables us to train two deep neural networks to represent two components of the solution in high-dimensional space. The curse of dimensionality is alleviated by using the Monte-Carlo method to discretize the unfitted Nitsche energy functional. We present several numerical examples to show the performance of the proposed method.