Deep unfitted Nitsche method for elliptic interface problems

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.

15 pages

References in corpus (4)

• PyTorch: An Imperative Style, High-Performance Deep Learning Library
• Convergence Rate Analysis for Deep Ritz Method
• A Priori Generalization Analysis of the Deep Ritz Method for Solving High Dimensional Elliptic Equations
• Unfitted Nitsche's method for computing band structures in phononic crystals with impurities

Cited by in corpus (4)

• A cusp-capturing PINN for elliptic interface problems
• A Shallow Ritz Method for Elliptic Problems with Singular Sources
• An efficient neural-network and finite-difference hybrid method for elliptic interface problems with applications
• A Compact Coupling Interface Method with Accurate Gradient Approximation for Elliptic Interface Problems