Open AccessAn immersed boundary neural network for solving elliptic equations with singular forces on arbitrary domains
Author(s) -
Reymundo Itzá Balam,
Francisco J. Hernandez-Lopez,
Joel Antonio Trejo-Sánchez,
Miguel Uh Zapata
Publication year - 2021
Publication title -
mathematical biosciences and engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.451
H-Index - 45
eISSN - 1551-0018
pISSN - 1547-1063
DOI - 10.3934/mbe.2021002
Subject(s) - singularity , artificial neural network , discontinuity (linguistics) , collocation (remote sensing) , boundary (topology) , boundary value problem , mathematical analysis , mathematics , computer science , artificial intelligence , machine learning
In this paper, we present a deep learning framework for solving two-dimensional elliptic equations with singular forces on arbitrary domains. This work follows the ideas of the physical-inform neural networks to approximate the solutions and the immersed boundary method to deal with the singularity on an interface. Numerical simulations of elliptic equations with regular solutions are initially analyzed in order to deeply investigate the performance of such methods on rectangular and irregular domains. We study the deep neural network solutions for different number of training and collocation points as well as different neural network architectures. The accuracy is also compared with standard schemes based on finite differences. In the case of singular forces, the analytical solution is continuous but the normal derivative on the interface has a discontinuity. This discontinuity is incorporated into the equations as a source term with a delta function which is approximated using a Peskin's approach. The performance of the proposed method is analyzed for different interface shapes and domains. Results demonstrate that the immersed boundary neural network can approximate accurately the analytical solution for elliptic problems with and without singularity.