Deep unfitted Nitsche method for elliptic interface problems

TL;DR

This paper introduces a deep neural network-based unfitted Nitsche method for high-dimensional elliptic interface problems with high contrast, effectively capturing discontinuities and alleviating the curse of dimensionality.

Contribution

It develops a novel deep unfitted Nitsche approach reformulating the problem as an energy minimization with two neural networks, addressing high-dimensional challenges.

Findings

Successfully captures solution discontinuities at interfaces.

Demonstrates effectiveness in high-dimensional problems.

Uses Monte-Carlo discretization to handle high-dimensional integrals.

Abstract

This paper proposes 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 performance of the proposed method.