@article{Azam2026, 
author = {Muhammad Azam and Dalal Alhwikem and Naseer Ullah},
title = {Hybrid PINN-IFE approach for solving transmission problems in circular interface domains},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {4},
pages = {10226-10256},
keywords = {physics-informed neural networks (PINNs), interface problems, hybrid methods, machine learning, transmission problems, wave propagation},
url = {https://www.sciopen.com/article/10.3934/math.2026423},
doi = {10.3934/math.2026423},
abstract = {In this study, we proposed a hybrid methodology combining Physics-Informed Neural Networks (PINNs) and Immersed Finite Element (IFE) methods to address transmission problems in complex geometries, with a focus on Helmholtz-type equations. The technique addressed the challenge of solving wave equations in domains with circular interfaces, where material properties differ across the interface. The hybrid model leverages the strengths of PINNs to enforce the governing physical equations and IFE to provide a coarse initial solution, which is then corrected by the neural network using a signed-distance function to the interface. This correction was trained on a combination of supervised loss from data, physics-informed residual predictions, and interface conditions. Numerical experiments demonstrated high precision of the proposed technique when compared with manufactured exact solutions, achieving low error levels in both subdomains. High-frequency tests at    ω  =  100 further validated the method's accuracy and robustness across material configurations including normal and inverted contrast cases. Its mesh-free character provides flexibility and versatility for a wide range of transmission problems in computational physics, making it a promising method for solving interface-related issues in wave propagation.}
}