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

Area of Science:

• Computational Mathematics
• Numerical Analysis
• Scientific Computing

Background:

• Elliptic equations are fundamental in modeling various physical phenomena.
• Solving these equations on arbitrary domains, especially with singularities, presents significant computational challenges.
• Existing numerical methods often struggle with accuracy and efficiency for complex geometries and singular forces.

Purpose of the Study:

• To develop a novel deep learning framework for solving two-dimensional elliptic equations.
• To address the challenges posed by singular forces and arbitrary domains using an immersed boundary approach.
• To evaluate the accuracy and performance of the proposed deep learning method compared to traditional schemes.

Main Methods:

• Utilizing physics-informed neural networks (PINNs) to approximate the solution of elliptic equations.
• Integrating the immersed boundary (IB) method to handle singularities at interfaces.
• Approximating delta functions for singular source terms using Peskin's approach.
• Investigating the impact of training parameters and network architectures on solution accuracy.

Main Results:

• The immersed boundary neural network (IBNN) framework demonstrates high accuracy in approximating solutions for elliptic equations with regular solutions on both rectangular and irregular domains.
• The method effectively handles singular forces by incorporating interface discontinuity into the equations as a source term.
• Performance analysis across various interface shapes and domains confirms the robustness and accuracy of the IBNN approach.
• Comparisons with finite difference methods show competitive or superior accuracy.

Conclusions:

• The proposed immersed boundary neural network is a powerful and accurate tool for solving two-dimensional elliptic equations, including those with singular forces and on complex domains.
• This framework offers a promising alternative to traditional numerical methods, particularly for problems with intricate geometries and localized singularities.
• The study validates the efficacy of combining deep learning with immersed boundary techniques for advanced scientific computing applications.