Physics-informed Kolmogorov–Arnold networks: Investigating architectures and hyperparameter impacts for solving Navier–Stokes equations

In recent years, physics-informed neural networks have demonstrated remarkable potential in solving partial differential equations (PDEs). Typically constructed using multilayer perceptrons (MLPs), these networks integrate physical laws into their training process, enabling solutions for both forward and inverse problems. Recently, Kolmogorov–Arnold Networks (KANs) have emerged as a promising alternative to MLPs due to their superior interpretability and accuracy in small-scale tasks. In this study, we propose a Physics-Informed Kolmogorov–Arnold Network (PI-KAN) model to solve forward problems of the Navier–Stokes equations, a fundamental system in fluid mechanics known for its nonlinearity and complexity. We systematically investigate the effects of different network architectures, hyperparameters, and collocation point distributions on the accuracy and convergence of PI-KANs. We also conducted a comparative study between PI-KANs and MLP-based PINNs to contrast the characteristics of both neural networks in solving the Navier–Stokes equations. Specifically, we analyze the information bottleneck phenomenon in multi-output KANs and propose methods to address it by modifying hidden-layer configurations. Furthermore, we explore the impact of random seed initialization on training outcomes and evaluate the efficacy of a pruning-based approach for network optimization. Our results demonstrate that PI-KANs achieve high prediction accuracy for Navier–Stokes equations with well-designed architectures, the mean squared error between the predicted velocity and the true velocity in our constructed network has reached the order of 10−5. Notably, uniform hidden-layer configurations yield optimal performance, while the balance between PDE and boundary condition losses plays a crucial role in achieving robust solutions. This study provides valuable insights into the design and implementation of PI-KANs for solving complex nonlinear PDEs, paving the way for broader applications in computational fluid dynamics and related fields.