Domain Decomposition Learning Methods for Elliptic Problems

Qi Sun¹, Xuejun Xu^1,2 and Haotian Yi¹

Paper: https://arxiv.org/abs/2207.10358

Abstract: Based on a direct transmission of Dirichlet and Neumann traces along subdomain interfaces, neural networks have already been employed as subproblem solvers in certain overlapping and non-overlapping methods. However, the boundary penalty treatment often leads to a tendency for the network solution and its derivatives to furnish more precision inside the domain, rather than at the boundary, thereby motivating the exploration of a variational approach for enforcing flux transmisssion with increased accuracy. In this study, a novel learning approach, i.e., the compensated deep Ritz method using neural network extension operators, is proposed to construct effective learning algorithms for realizing non-overlapping domain decomposition methods even in the presence of inaccurate interface conditions.

_{¹School of Mathematical Sciences, Tongji University, Shanghai 200092, China, TX
²Institute of Computational Mathematics, AMSS, Chinese Academy of Sciences, Beijing 100190, China, MD}

Preview of results

Network solutions and error profiles for Poisson problem using DN-PINNs and DNLA (PINNs), with fine-tuned hyperparameters.

Requirements

Code was implemented in python 3.7 with the following package versions:

pytorch version = 1.8.1 + cu111
tensorflow version = 2.8.0

and Matlab 2023b was used for visualization.

Citation

@article{sun2022domain,
  title={Domain Decomposition Learning Methods for Solving Elliptic Problems},
  author={Sun, Qi and Xu, Xuejun and Yi, Haotian},
  journal={arXiv preprint arXiv:2207.10358},
  year={2022}
        }