Afrah Farea

Afrah Farea, Saiful Khan, Mustafa Serdar Celebi

Physics-informed neural networks (PINNs) have emerged as promising methods for solving partial differential equations (PDEs) by embedding physical laws within neural architectures. However, these classical approaches often require a large number of parameters to achieve reasonable accuracy, particularly for complex PDEs. In this paper, we present a quantum-classical physics-informed neural network (QCPINN) that combines quantum and classical components, allowing us to solve PDEs with significantly fewer parameters while maintaining comparable accuracy and convergence to classical PINNs. We systematically evaluated two quantum circuit architectures across various configurations on five benchmark PDEs to identify optimal QCPINN designs. Our results demonstrate that the QCPINN achieves stable convergence and comparable accuracy while using only 10-30% of the trainable parameters required by classical PINNs. This approach also results in a significant reduction in the relative L_2 error for Helmholtz, Klein-Gordon, and Convection-diffusion equations, with a reduction ranging from 4% to 64% across various fields. These findings demonstrate the potential of parameter efficiency and solution accuracy in physics-informed machine learning, allowing for a substantial decrease in model complexity without compromising solution quality.QCPINN presents a promising pathway to address the computational challenges associated with solving PDEs.

Afrah Farea, Mustafa Serdar Celebi

Physics-Informed Neural Networks (PINNs) have emerged as a promising approach for solving Partial Differential Equations (PDEs). However, they face challenges related to spectral bias (the tendency to learn low-frequency components while struggling with high-frequency features) and unstable convergence dynamics (mainly stemming from the multi-objective nature of the PINN loss function). These limitations impact their accuracy for problems involving rapid oscillations, sharp gradients, and complex boundary behaviors. We systematically investigate learnable activation functions as a solution to these challenges, comparing Multilayer Perceptrons (MLPs) using fixed and learnable activation functions against Kolmogorov-Arnold Networks (KANs) that employ learnable basis functions. Our evaluation spans diverse PDE types, including linear and non-linear wave problems, mixed-physics systems, and fluid dynamics. Using empirical Neural Tangent Kernel (NTK) analysis and Hessian eigenvalue decomposition, we assess spectral bias and convergence stability of the models. Our results reveal a trade-off between expressivity and training convergence stability. While learnable activation functions work well in simpler architectures, they encounter scalability issues in complex networks due to the higher functional dimensionality. Counterintuitively, we find that low spectral bias alone does not guarantee better accuracy, as functions with broader NTK eigenvalue spectra may exhibit convergence instability. We demonstrate that activation function selection remains inherently problem-specific, with different bases showing distinct advantages for particular PDE characteristics. We believe these insights will help in the design of more robust neural PDE solvers.

Afrah Farea, Saiful Khan, Reza Daryani et al.

Physics-informed neural networks (PINNs) have emerged as a promising approach for solving complex fluid dynamics problems, yet their application to fluid-structure interaction (FSI) problems with moving boundaries remains largely unexplored. This work addresses the critical challenge of modeling FSI systems with deformable interfaces, where traditional unified PINN architectures struggle to capture the distinct physics governing fluid and structural domains simultaneously. We present an innovative Eulerian-Lagrangian PINN architecture that integrates immersed boundary method (IBM) principles to solve FSI problems with moving boundary conditions. Our approach fundamentally departs from conventional unified architectures by introducing domain-specific neural networks: an Eulerian network for fluid dynamics and a Lagrangian network for structural interfaces, coupled through physics-based constraints. Additionally, we incorporate learnable B-spline activation functions with SiLU to capture both localized high-gradient features near interfaces and global flow patterns. Empirical studies on a 2D cavity flow problem involving a moving solid structure show that while baseline unified PINNs achieve reasonable velocity predictions, they suffer from substantial pressure errors (12.9%) in structural regions. Our Eulerian-Lagrangian architecture with learnable activations (EL-L) achieves better performance across all metrics, improving accuracy by 24.1-91.4% and particularly reducing pressure errors from 12.9% to 2.39%. These results demonstrate that domain decomposition aligned with physical principles, combined with locality-aware activation functions, is essential for accurate FSI modeling within the PINN framework.