[論文レビュー] Physics-informed Kolmogorov-Arnold Network with Chebyshev Polynomials for Fluid Mechanics

Chunyu Guo, Lu Sun+3 more|arXiv (Cornell University)|Nov 7, 2024

Computational Physics and Python Applications被引用数 7

ひとこと要約

The paper introduces ChebPIKAN, a Chebyshev-based Chebyshev-augmented Kolmogorov-Arnold Network with physics-informed losses to solve various PDEs in fluid mechanics, showing improved accuracy and reduced overfitting over standard KAN and PINN approaches.

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 have improved 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.

研究の動機と目的

• Motivate fast, accurate PDE solving in fluid dynamics with limited data.
• Integrate physical constraints into Kolmogorov-Arnold networks to improve generalization.
• Develop a Chebyshev-basis KAN architecture (ChebPIKAN) to enhance interpolation and fidelity of PDE solutions.
• Evaluate ChebPIKAN on multiple PDEs including Allen-Cahn, Burgers, Helmholtz, Kovasznay flow, and Navier–Stokes.

提案手法

• Use Kolmogorov-Arnold networks (KAN) as the base architecture, leveraging the Kolmogorov-Arnold representation to learn univariate spline-based activations.
• Replace/augment standard activations with Chebyshev polynomial basis functions to enforce orthogonality and reduce multicollinearity in fits.
• Incorporate physics-informed loss terms by adding PDE residuals to the data loss, creating a Loss = Loss_data + lambda * Loss_physics objective.
• Apply the framework to 1D Allen-Cahn and Burgers equations, and 2D Helmholtz, Kovasznay flow, and Navier–Stokes equations, with varying layer depths.
• Investigate network performance across shallow to deep configurations, using Adam optimization and PDE-residual based training signals.

実験結果

リサーチクエスチョン

• RQ1 Can ChebPIKAN improve PDE solution accuracy over standard KAN and PINN by incorporating physical information into the KAN framework?
• RQ2 Does adding Chebyshev-based basis functions enhance interpolation and reduce overfitting in KAN for fluid-mechanics PDEs?
• RQ3 How does network depth affect performance for ChebPIKAN across different fluid dynamics equations?
• RQ4 What is the impact on extrapolation ability when physical constraints are embedded in KAN?

主な発見

• ChebPIKAN consistently achieves lower residuals than KAN across tested equations, with four hidden layers achieving particularly strong performance for Allen-Cahn.
• ChebPIKAN substantially mitigates overfitting observed in KAN, especially at higher network depths, by embedding physical information.
• For Burgers and Helmholtz problems, ChebPIKAN demonstrates robustness and improved prediction accuracy in regions with high gradients or complexity.
• In the Navier–Stokes experiments, deeper architectures with ChebPIKAN show favorable learning behavior and PDE-consistent losses, suggesting better generalization and fast, reliable predictions.