A mesh-free method using piecewise deep neural network for elliptic interface problems

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-d...

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Vydáno v:Journal of computational and applied mathematics Ročník 412; s. 114358
Hlavní autoři: He, Cuiyu, Hu, Xiaozhe, Mu, Lin
Médium: Journal Article
Jazyk:angličtina
Vydáno: Elsevier B.V 01.10.2022
Témata:
Adaptive method
DNN
Interface problems
Least-square method
Mesh-free
Neural networks
Least-square method
Neural networks
DNN
Adaptive method
Mesh-free
Interface problems
ISSN:0377-0427, 1879-1778
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Shrnutí: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.
ISSN:0377-0427
1879-1778
DOI:10.1016/j.cam.2022.114358