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.

翻译：Kolmogorov-Arnold网络（KANs）近期在求解偏微分方程（PDEs）方面展现出潜力。然而，其原始形式在计算和内存方面开销较大，这促使了基于第一类切比雪夫多项式的KANs（Chebyshev1KANs）的提出。尽管Chebyshev1KANs的性能已优于原始KANs架构，但我们严格的理论分析表明，它们仍然存在秩崩溃问题，最终限制了其表达能力。为克服这些局限，我们通过集成具有可学习参数的小波激活多层感知器（MLPs）以及内部注意力机制来增强Chebyshev1KANs。我们证明，该设计保持了雅可比矩阵的满秩性，并能够逼近任意阶偏微分方程的解。此外，为缓解切比雪夫多项式基引入的损失不稳定与不平衡问题，我们外部引入了一种残差梯度注意力（RGA）机制，该机制根据各项损失项的梯度范数和残差大小动态地重新加权。通过联合利用内部与外部注意力，我们提出了AC-PKAN这一新颖架构，它增强了对弱监督物理信息神经网络（PINNs）的支持，并扩展了KANs的表达能力。在三个领域的九项基准任务上的实验结果表明，AC-PKAN始终优于或匹配PINNsFormer等最先进模型，确立了其作为在零数据或数据稀疏场景下解决复杂现实工程问题的高效工具的地位。代码将在论文被接受后公开。