KAN在物理约束模型中表现不稳定，尤其对复杂耦合项效果差。
Empirical Stability Analysis of Kolmogorov-Arnold Networks in Hard-Constrained Recurrent Physics-Informed Discovery
- 将KAN嵌入递归物理信息网络，测试其对振荡系统残差的恢复能力
- 小规模KAN在单变量多项式上表现尚可，但深层配置易失稳
- 对乘积项建模失败，不如传统MLP，暴露了KAN归纳偏置局限性
我们研究将柯尔莫哥洛夫-阿诺德网络（KAN）集成到硬约束递归物理信息架构（HRPINN）中，评估其在振荡系统中学习残差流形的保真度。基于柯尔莫哥洛夫-阿诺德表示定理及初步灰盒结果，假设KAN相比多层感知机（MLP）能更高效恢复未知项。通过初始敏感性分析配置敏感性、参数尺度和训练范式，发现尽管小型KAN在单变量多项式残差（杜芬系统）上具有竞争力，但在深层结构中表现出严重的超参数脆弱性，且在乘积项（范德波尔系统）上持续失效，总体性能普遍低于标准MLP。这些实证挑战揭示了原始KAN形式中加性归纳偏置在状态耦合中的局限性，为未来混合建模提供了初步实证依据。
原文摘要 · Abstract (English)
We investigate the integration of Kolmogorov-Arnold Networks (KANs) into hard-constrained recurrent physics-informed architectures (HRPINN) to evaluate the fidelity of learned residual manifolds in oscillatory systems. Motivated by the Kolmogorov-Arnold representation theorem and preliminary gray-box results, we hypothesized that KANs would enable efficient recovery of unknown terms compared to MLPs. Through initial sensitivity analysis on configuration sensitivity, parameter scale, and training paradigm, we found that while small KANs are competitive on univariate polynomial residuals (Duffing), they exhibit severe hyperparameter fragility, instability in deeper configurations, and consistent failure on multiplicative terms (Van der Pol), generally outperformed by standard MLPs. These empirical challenges highlight limitations of the additive inductive bias in the original KAN formulation for state coupling and provide preliminary empirical evidence of inductive bias limitations for future hybrid modeling.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。