Physics-Informed Kolmogorov-Arnold networks for viscoelastic fluid equations

Abstract

Kolmogorov-Arnold Networks (KANs), inspired by the Kolmogorov Arnold representation theorem, provide an interpretable alternative to multilayer perceptrons (MLPs) by using learnable activation functions on edges rather than fixed node activations. We propose a Physics-Informed Kolmogorov-Arnold Network (PI-KAN) framework for solving forward problem of viscoelastic fluid equations, which arise in many complex fluid dynamics applications and are characterized by strong nonlinear coupling between fluid fields. For viscoelastic fluid equations, we adopt the generalized hydrodynamic model, which is well established in the field of dusty plasma. To evaluate the performance of the proposed framework for viscoelastic fluid, we consider benchmark problem based on the Taylor-Green (TG) flow and a modified Taylor-Green flow. We systematically investigate the effects of different network architectures, hyperparameters, and collocation point distributions on the accuracy and convergence behavior of PI-KANs for the range of viscoelastic parameter ($\tau_m = 1$--$20$). We also study the impact of random seed initialization on training outcomes. The obtained results provide useful guidance for the design and implementation of physics-informed Kolmogorov-Arnold networks (PI-KANs) in solving viscoelastic fluid equations