# Dirichlet-Neumann learning algorithm for solving elliptic interface problems

arXiv: https://arxiv.org/abs/2301.07361 · HTML: https://mlsanity.com/paper/2301.07361/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 63 references that are in the map.
- First posted on arXiv 18 Jan 2023.

## Builds on (most cited first, 10 of 10)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Mitigating Propagation Failures in Physics-informed Neural Networks using Retain-Resample-Release (R3) Sampling](https://mlsanity.com/paper/2207.02338/) (2022)
- [When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization?](https://mlsanity.com/paper/2109.09444/) (2021)
- [Augmented Physics-Informed Neural Networks (APINNs): A gating network-based soft domain decomposition methodology](https://mlsanity.com/paper/2211.08939/) (2022)
- [Recipes for when Physics Fails: Recovering Robust Learning of Physics Informed Neural Networks](https://mlsanity.com/paper/2110.13330/) (2021)
- [A cusp-capturing PINN for elliptic interface problems](https://mlsanity.com/paper/2210.08424/) (2022)
- [A Discussion on Solving Partial Differential Equations using Neural Networks](https://mlsanity.com/paper/1904.07200/) (2019)
- [Learning Interface Conditions in Domain Decomposition Solvers](https://mlsanity.com/paper/2205.09833/) (2022)
- [A coarse space acceleration of deep-DDM](https://mlsanity.com/paper/2112.03732/) (2021)
- [An efficient neural-network and finite-difference hybrid method for elliptic interface problems with applications](https://mlsanity.com/paper/2210.05523/) (2022)

## Abstract (arXiv)

Non-overlapping domain decomposition methods are natural for solving interface problems arising from various disciplines, however, the numerical simulation requires technical analysis and is often available only with the use of high-quality grids, thereby impeding their use in more complicated situations. To remove the burden of mesh generation and to effectively tackle with the interface jump conditions, a novel mesh-free scheme, i.e., Dirichlet-Neumann learning algorithm, is proposed in this work to solve the benchmark elliptic interface problem with high-contrast coefficients as well as irregular interfaces. By resorting to the variational principle, we carry out a rigorous error analysis to evaluate the discrepancy caused by the boundary penalty treatment for each decomposed subproblem, which paves the way for realizing the Dirichlet-Neumann algorithm using neural network extension operators. The effectiveness and robustness of our proposed methods are demonstrated experimentally through a series of elliptic interface problems, achieving better performance over other alternatives especially in the presence of erroneous flux prediction at interface.