J-PIKAN: A physics-informed KAN network based on Jacobi orthogonal polynomials for solving fluid dynamics

Traditional Physics-Informed Neural Networks (PINNs) based on multilayer perceptrons face optimization difficulties, spectral bias, and limited parameter efficiency when solving complex fluid dynamics problems. This work addresses these limitations by developing J-PIKAN, a physics-informed Kolmogorov-Arnold Network based on Jacobi orthogonal polynomials.We present a systematic comparison of different basis functions (Jacobi polynomials with various α,β parameters, Chebyshev, Legendre, Hermite, Fourier, B-spline, and Taylor) across five representative fluid dynamics benchmarks. Our comprehensive analysis reveals that Jacobi polynomials consistently achieve superior performance, delivering 1–2 orders of magnitude improvement in solution accuracy compared to baseline MLPs across different equation types. Through Hessian eigenvalue analysis, we demonstrate that J-PIKAN exhibits more favorable optimization characteristics with reduced numerical ill-conditioning during training. For high Reynolds number lid-driven cavity flows, J-PIKAN maintains superior accuracy while requiring only 50% of the parameters compared to basic MLPs. J-PIKAN offers a promising framework for developing efficient and reliable deep learning-based solvers for fluid dynamics, demonstrating significant improvements in both accuracy and parameter efficiency while addressing numerical stability challenges associated with polynomial-based networks. Code will be made available at https://github.com/xgxgnpu/J-PIKAN upon acceptance of the paper.