A Hybrid Iterative Deep Ritz Method for Elliptic Interface Problems

2026-09-22

AI Summary

提出了一种混合迭代深度Ritz方法（H-IDRM）来解决椭圆型界面问题，利用水平集神经网络架构处理分段光滑解和复杂几何界面。

Abstract

In this work, we propose a hybrid iterative deep Ritz method (H-IDRM) for a class of interface problems for second-order elliptic operators. It is based on a new mixed formulation of the problem and involves solving a sequence of convex minimization problems. We employ a level-set neural network architecture, featuring a level-set representation of the interface, to accommodate the piecewise smoothness of the solution and the flux. The approach involves only volumetric representations instead of duality pairing on the interface and avoids explicit interface sampling that is inconvenient for complex interface geometries. Further, we present an analysis of the method, including the errors arising from the neural network approximation, Monte Carlo approximation, iterative scheme, and penalty parameters. Numerical experiments indicate that the H-IDRM outperforms existing neural solvers on problems with high-dimensional domains, intricate interface geometries, and lower subdomain regularity.

Problem

• Elliptic Interface Problems
• High-dimensional Domains
• Intricate Interface Geometries
• Subdomain Regularity

Innovation

• Hybrid Iterative Deep Ritz Method
• Level-set Neural Network Architecture
• Convex Minimization Problems
• Volumetric Representations