AC-PKAN: Attention-Enhanced and Chebyshev Polynomial-Based Physics-Informed Kolmogorov–Arnold Networks

Hangwei Zhang, Yan Wang

Keywords: Physics-Informed Neural Networks, Kolmogorov–Arnold Networks, Attention Mechanism, PDEs, Chebyshev Polynomials

TL;DR: We introduce AC-PKAN, a novel framework that enhances Physics-Informed Neural Networks (PINNs) with Chebyshev polynomials and attention mechanisms to improve efficiency and accuracy in solving complex partial differential equations.

Abstract: This paper introduces AC-PKAN, an advanced framework for Physics-Informed Neural Networks (PINNs) that integrates Kolmogorov–Arnold Networks (KANs) with Chebyshev Type-I polynomials and incorporates both internal and external attention mechanisms. Traditional PINNs based on Multilayer Perceptrons (MLPs) encounter challenges when handling complex partial differential equations (PDEs) due to vanishing gradients, limited interpretability, and computational inefficiency. To address these issues, we enhance the model from both external and internal perspectives. Externally, we propose a novel Residual Gradient Attention (RGA) mechanism that dynamically adjusts loss term weights based on gradient norms and residuals, thereby mitigating gradient stiffness and residual imbalance. Internally, AC-PKAN employs point-wise Chebyshev polynomial-based KANs, wavelet-activated MLPs with learnable parameters, and internal attention mechanisms. These integrated components improve both training efficiency and prediction accuracy. We provide mathematical proofs demonstrating that AC-PKAN can theoretically solve any finite-order PDE. Experimental results from five benchmark tasks across three domains show that AC-PKAN consistently outperforms or matches state-of-the-art models such as PINNsFormer, establishing it as a highly effective tool for solving complex real-world engineering problems.

Primary Area: applications to physical sciences (physics, chemistry, biology, etc.)

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Submission Number: 2375