# Two-level deep domain decomposition method

arXiv: https://arxiv.org/abs/2408.12198 · HTML: https://mlsanity.com/paper/2408.12198/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 13 references that are in the map.
- First posted on arXiv 22 Aug 2024.

## Builds on (most cited first, 7 of 7)

- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks](https://mlsanity.com/paper/1901.06523/) (2019)
- [On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks](https://mlsanity.com/paper/2012.10047/) (2020)
- [Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equations](https://mlsanity.com/paper/2107.07871/) (2021)
- [Parallel Physics-Informed Neural Networks via Domain Decomposition](https://mlsanity.com/paper/2104.10013/) (2021)
- [D3M: A deep domain decomposition method for partial differential equations](https://mlsanity.com/paper/1909.12236/) (2019)
- [A Block-Coordinate Approach of Multi-level Optimization with an Application to Physics-Informed Neural Networks](https://mlsanity.com/paper/2305.14477/) (2023)

## Abstract (arXiv)

This study presents a two-level Deep Domain Decomposition Method (Deep-DDM) augmented with a coarse-level network for solving boundary value problems using physics-informed neural networks (PINNs). The addition of the coarse level network improves scalability and convergence rates compared to the single level method. Tested on a Poisson equation with Dirichlet boundary conditions, the two-level deep DDM demonstrates superior performance, maintaining efficient convergence regardless of the number of subdomains. This advance provides a more scalable and effective approach to solving complex partial differential equations with machine learning.