AC-PKAN：基于注意力机制和切比雪夫多项式的物理信息柯莫戈罗夫-阿诺德网络
AC-PKAN: Attention-Enhanced and Chebyshev Polynomial-Based Physics-Informed Kolmogorov-Arnold Networks
AI总结：
AC-PKAN通过结合注意力机制和切比雪夫多项式，提升物理信息神经网络在求解偏微分方程中的性能与稳定性。
摘要：
柯莫戈罗夫-阿诺德网络（KANs）最近在求解偏微分方程（PDEs）方面显示出潜力。然而，其原始公式在计算和内存消耗上都很高，促使引入基于切比雪夫类型I的KANs（Chebyshev1KANs）。尽管Chebyshev1KANs在性能上优于传统的KANs架构，但我们的严格理论分析表明，它们仍然存在秩坍缩问题，最终限制了其表达能力。为克服这些限制，我们通过将小波激活的多层感知机（MLP）与可学习参数和内部注意力机制相结合，增强Chebyshev1KANs。我们证明了这种设计能够保持满秩雅可比矩阵，并能够近似任意阶次的PDE解。此外，为了缓解切比雪夫多项式基底引入的损失不稳定性和不平衡问题，我们外部引入了残差梯度注意力（RGA）机制，该机制根据梯度范数和残差大小动态重新加权个体损失项。通过联合利用内部和外部注意力，我们提出了AC-PKAN，这是一种增强弱监督物理信息神经网络（PINNs）并扩展KANs表达能力的新架构。在三个领域中的九个基准任务上的实验结果表明，AC-PKAN在性能上一致优于或匹配了最先进的模型，如PINNsFormer，证明其在零数据或数据稀疏条件下解决复杂现实工程问题的有效性。代码将在接受后公开。
英文摘要：
Kolmogorov-Arnold Networks (KANs) have recently shown promise for solving partial differential equations (PDEs). Yet their original formulation is computationally and memory intensive, motivating the introduction of Chebyshev Type-I-based KANs (Chebyshev1KANs). Although Chebyshev1KANs have outperformed the vanilla KANs architecture, our rigorous theoretical analysis reveals that they still suffer from rank collapse, ultimately limiting their expressive capacity. To overcome these limitations, we enhance Chebyshev1KANs by integrating wavelet-activated MLPs with learnable parameters and an internal attention mechanism. We prove that this design preserves a full-rank Jacobian and is capable of approximating solutions to PDEs of arbitrary order. Furthermore, to alleviate the loss instability and imbalance introduced by the Chebyshev polynomial basis, we externally incorporate a Residual Gradient Attention (RGA) mechanism that dynamically re-weights individual loss terms according to their gradient norms and residual magnitudes. By jointly leveraging internal and external attention, we present AC-PKAN, a novel architecture that constitutes an enhancement to weakly supervised Physics-Informed Neural Networks (PINNs) and extends the expressive power of KANs. Experimental results from nine 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 in zero-data or data-sparse regimes. The code will be made publicly available upon acceptance.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
