A Hybrid Iterative Deep Ritz Method for Elliptic Interface Problems
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.
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BibTeX
@misc{hu2026hybrid,
title = {{A Hybrid Iterative Deep Ritz Method for Elliptic Interface Problems}},
author = {Tianhao Hu and Bangti Jin and Fengru Wang and Yifeng Xu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27325},
eprint = {2609.27325},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27325},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A Hybrid Iterative Deep Ritz Method for Elliptic Interface Problems
AU - Tianhao Hu
AU - Bangti Jin
AU - Fengru Wang
AU - Yifeng Xu
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27325
UR - https://arxiv.org/abs/2609.27325
AB - 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.
ER -