Physics-informed Kolmogorov-Arnold Network with Chebyshev Polynomials for Fluid Mechanics (2411.04516v2)

Abstract: Solving partial differential equations (PDEs) is essential in scientific forecasting and fluid dynamics. Traditional approaches often incur expensive computational costs and trade-offs in efficiency and accuracy. Recent deep neural networks improve the accuracy but require high-quality training data. Physics-informed neural networks (PINNs) effectively integrate physical laws to reduce the data reliance in limited sample scenarios. A novel machine-learning framework, Chebyshev physics-informed Kolmogorov--Arnold network (ChebPIKAN), is proposed to integrate the robust architectures of Kolmogorov--Arnold networks (KAN) with physical constraints to enhance the calculation accuracy of PDEs for fluid mechanics. We study the fundamentals of KAN, take advantage of the orthogonality of Chebyshev polynomial basis functions in spline fitting, and integrate physics-informed loss functions that are tailored to specific PDEs in fluid dynamics, including Allen--Cahn equation, nonlinear Burgers equation, Helmholtz equations, Kovasznay flow, cylinder wake flow, and cavity flow. Extensive experiments demonstrate that the proposed ChebPIKAN model significantly outperforms the standard KAN architecture in solving various PDEs by effectively embedding essential physical information. These results indicate that augmenting KAN with physical constraints can alleviate the overfitting issues of KAN and improve the extrapolation performance. Consequently, this study highlights the potential of ChebPIKAN as a powerful tool in computational fluid dynamics and propose a path toward fast and reliable predictions in fluid mechanics and beyond.