Hybrid PINN-IFE approach for solving transmission problems in circular interface domains

Muhammad Azam; Dalal Alhwikem; Naseer Ullah; Muhammad Azam; Dalal Alhwikem; Naseer Ullah

doi:10.3934/math.2026423

AIMS Mathematics

2026, Volume 11, Issue 4: 10226-10256. doi: 10.3934/math.2026423

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Hybrid PINN-IFE approach for solving transmission problems in circular interface domains

1.
Key Laboratory of Urban Security and Disaster Engineering of the Ministry of Education, Beijing University of Technology, Beijing 100124, China
2.
Department of Mathematics, College of Science, Qassim University, Buraydah 52571, Saudi Arabia

Received: 13 December 2025 Revised: 26 February 2026 Accepted: 09 March 2026 Published: 15 April 2026

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 $ \omega = 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.
- physics-informed neural networks (PINNs),
- interface problems,
- hybrid methods,
- machine learning,
- transmission problems,
- wave propagation
Citation: Muhammad Azam, Dalal Alhwikem, Naseer Ullah. Hybrid PINN-IFE approach for solving transmission problems in circular interface domains[J]. AIMS Mathematics, 2026, 11(4): 10226-10256. doi: 10.3934/math.2026423

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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 $ \omega = 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.

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