# A Hybrid Iterative Deep Ritz Method for Elliptic Interface Problems

arXiv: https://arxiv.org/abs/2609.27325 · HTML: https://mlsanity.com/paper/2609.27325/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 40 references that are in the map.
- First posted on arXiv 23 Sep 2026.

## Builds on (most cited first, 6 of 6)

- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [A Discontinuity Capturing Shallow Neural Network for Elliptic Interface Problems](https://mlsanity.com/paper/2106.05587/) (2021)
- [A cusp-capturing PINN for elliptic interface problems](https://mlsanity.com/paper/2210.08424/) (2022)
- [DRM Revisited: A Complete Error Analysis](https://mlsanity.com/paper/2407.09032/) (2024)
- [Point Source Identification Using Singularity Enriched Neural Networks](https://mlsanity.com/paper/2408.09143/) (2024)
- [An Iterative Deep Ritz Method for Monotone Elliptic Problems](https://mlsanity.com/paper/2501.15186/) (2025)

## Abstract (arXiv)

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.