A Novel Hybrid Boundary Element—Physics Informed Neural Network Method for Numerical Solutions in Electromagnetics

This repository is the official implementation of the paper:

“A Novel Hybrid Boundary Element—Physics Informed Neural Network Method for Numerical Solutions in Electromagnetics”
Published in IEEE Access, 2024
https://ieeexplore.ieee.org/document/10755077

It provides a hybrid computational framework that combines the Boundary Element Method (BEM) and Physics-Informed Neural Networks (PINNs) for solving electromagnetic problems governed by the Laplace equation.

The repository reproduces and extends the methodology proposed in the paper, including domain decomposition and coupling via interface boundary conditions.

This work introduces a hybrid BEM–PINN approach for numerical solutions in electromagnetics:

• PINN is used to solve the PDE in part of the domain
• BEM is applied to the remaining subdomain
• The two methods are coupled through interface conditions

This approach leverages:

• the flexibility of neural networks
• the efficiency of boundary-based numerical methods

to solve partial differential equations in complex domains.

Boundary Element Method (BEM), Physics-Informed Neural Networks (PINNs), Electromagnetics, Laplace Equation, Scientific Machine Learning, Domain Decomposition

🔹 v1.0.0 — Homogeneous L-Shaped Domain Folder: Homogeneous L-Shaped Domain (V1.0.0) Baseline implementation of the hybrid BEM–PINN framework Solves the Laplace equation on a homogeneous L-shaped domain Domain decomposition: Left subdomain → solved using PINN Right-bottom subdomain → solved using BEM Coupling enforced through interface boundary conditions 👉 This version validates the methodology presented in the paper.

BEM–PINN coupling on L-shaped domain

🔹 v2.0.0 — Multi-Material Nested Square Domains Extension from homogeneous to heterogeneous (multi-material) domains Support for nested square geometries with material interfaces Coupled BEM–PINN framework with interface continuity enforcement Continuity of scalar potential and normal flux across interfaces Improved modular architecture for multi-domain simulations Validation on heterogeneous benchmark problems 👉 This version validates the methodology presented in the paper.

BEM–PINN coupling on Nested Square Domains

• MATLAB
• Deep Learning Toolbox
• Optimization Toolbox
• Parallel Computing Toolbox

Each version produces:

• Scalar potential field
• Gradient fields
• BEM reconstruction
• Combined contour visualization

The scientific foundation of this repository is described in:

Barmada, S., Dodge, S., Tucci, M., Formisano, A., Di Barba, P., & Mognaschi, M. E.
“A Novel Hybrid Boundary Element—Physics Informed Neural Network Method for Numerical Solutions in Electromagnetics.”
IEEE Access, 2024
https://ieeexplore.ieee.org/document/10755077

📌 This repository is the official codebase accompanying the above publication.

If you use this work, please cite:

@article{barmada2024hybrid,
  title={A Novel Hybrid Boundary Element—Physics Informed Neural Network Method for Numerical Solutions in Electromagnetics},
  author={Barmada, Sami and Dodge, Shayan and Tucci, Mauro and Formisano, Alessandro and Di Barba, Paolo and Mognaschi, Maria Evelina},
  journal={IEEE Access},
  volume={12},
  pages={171444--171457},
  year={2024},
  publisher={IEEE}
}