School of Mathematical and Statistical Sciences Faculty Publications and Presentations
Document Type
Article
Publication Date
10-2022
Abstract
In this paper, we propose a novel mesh-free numerical method for solving the elliptic interface problems based on deep learning. We approximate the solution by the neural networks and, since the solution may change dramatically across the interface, we employ different neural networks for each sub-domain. By reformulating the interface problem as a least-squares problem, we discretize the objective function using mean squared error via sampling and solve the proposed deep least-squares method by standard training algorithms such as stochastic gradient descent. The discretized objective function utilizes only the point-wise information on the sampling points and thus no underlying mesh is required. Doing this circumvents the challenging meshing procedure as well as the numerical integration on the complex interfaces. To improve the computational efficiency for more challenging problems, we further design an adaptive sampling strategy based on the residual of the least-squares function and propose an adaptive algorithm. Finally, we present several numerical experiments in both 2D and 3D to show the flexibility, effectiveness, and accuracy of the proposed deep least-square method for solving interface problems.
Recommended Citation
He, Cuiyu, Xiaozhe Hu, and Lin Mu. "A mesh-free method using piecewise deep neural network for elliptic interface problems." Journal of Computational and Applied Mathematics 412 (2022): 114358. https://doi.org/10.1016/j.cam.2022.114358
Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial-No Derivative Works 4.0 International License.
Publication Title
Journal of Computational and Applied Mathematics
DOI
10.1016/j.cam.2022.114358
Comments
Original published version available at https://doi.org/10.1016/j.cam.2022.114358