An immersed boundary neural network for solving elliptic equations with singular forces on arbitrary domains

Reymundo Itzá Balam; Francisco Hernandez-Lopez; Joel Trejo-Sánchez; Miguel Uh Zapata; Reymundo Itzá Balam; Francisco Hernandez-Lopez; Joel Trejo-Sánchez; Miguel Uh Zapata

doi:10.3934/mbe.2021002

Mathematical Biosciences and Engineering

2021, Volume 18, Issue 1: 22-56. doi: 10.3934/mbe.2021002

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An immersed boundary neural network for solving elliptic equations with singular forces on arbitrary domains

1.
Centro de Investigación en Matemáticas A.C, CIMAT-Mérida, México
2.
Consejo Nacional de Ciencia y Tecnología, CONACYT, México

Received: 09 August 2020 Accepted: 02 November 2020 Published: 18 November 2020

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.
- elliptic equation,
- immersed boundary method,
- singular forces,
- neural networks,
- deep learning,
- continuous time model,
- interface,
- two-phase flow
Citation: Reymundo Itzá Balam, Francisco Hernandez-Lopez, Joel Trejo-Sánchez, Miguel Uh Zapata. An immersed boundary neural network for solving elliptic equations with singular forces on arbitrary domains[J]. Mathematical Biosciences and Engineering, 2021, 18(1): 22-56. doi: 10.3934/mbe.2021002

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Abstract

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.

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