Kolmogorov-Arnold网络在硬约束递归物理信息发现中的经验稳定性分析
Empirical Stability Analysis of Kolmogorov-Arnold Networks in Hard-Constrained Recurrent Physics-Informed Discovery
AI总结：
本文研究了Kolmogorov-Arnold网络在硬约束递归物理信息发现中的稳定性，发现其在处理乘法项时表现欠佳，且存在超参数脆弱性问题，表明原有加法归纳偏置的局限性。
摘要：
我们研究了将Kolmogorov-Arnold网络（KANs）整合到硬约束递归物理信息架构（HRPINN）中，以评估在振荡系统中学习残差流形的保真度。受Kolmogorov-Arnold表示定理和初步灰盒结果的启发，我们假设KANs相比MLP能够更高效地恢复未知项。通过初始的灵敏度分析，我们发现虽然小规模KAN在单变量多项式残差（Duffing）上具有竞争力，但在更深层次的配置中表现出严重的超参数脆弱性、不稳定性和在乘法项（Van der Pol）上的一致失败，总体上不如标准MLP表现优异。这些经验挑战突显了原始KAN公式中加法归纳偏置在状态耦合中的局限性，并为未来混合建模提供了初步的经验证据。
英文摘要：
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.
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