发表机构
Clarkson University（克拉克森大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出Higher-Order-KANDy，利用延迟和导数嵌入及KANDy层，无需候选库即可从含噪数据中学习高阶动力系统方程，在噪声高达7.5%时仍能准确选择活跃项。
AI 中文摘要
从数据中推导动力系统的控制方程对于许多科学领域的预测和控制至关重要。然而，存在两个挑战：将高阶方程降为一阶形式时候选项的扩展，以及估计导数时引入的噪声。我们提出Higher-Order-KANDy，该方法通过使用原始数据的延迟和导数（“jet”）嵌入，并将导数阶Kolmogorov-Arnold网络用于动力学（KANDy）层堆叠到机器学习架构中，同时解决这两个问题。与SINDy和WSINDy在含噪数据上的比较表明，Higher-Order-KANDy无需候选库即可学习控制方程（例如，无需预先指定任何项即可恢复完整周期的$-\sin(u)$），天然适用于高阶系统，并在噪声增加时保持正确的项结构，在噪声水平高达7.5%时，在每个种子上选择精确的活跃集，而oracle弱形式回归则会出现虚假项，尽管oracle仍然是更准确的系数估计器。我们利用延迟和微分嵌入之间的关系，以及Kolmogorov-Arnold表示定理，构建延迟到jet的映射，并将其作为Kolmogorov-Arnold网络学习，以保留导数结构并提供控制方程中出现的有意义项。在我们的合成基准测试中，该构造在粗采样和噪声存在的情况下恢复了方程。
英文摘要
The ability to derive the governing equations of a dynamical system from data is essential for prediction and control across many scientific fields. Two challenges, however, are the expansion of candidate terms when reducing higher-order equations to first-order form and noise from estimating derivatives. We propose Higher-Order-KANDy, which addresses both problems by using delay and derivative ("jet") embeddings of the original data with derivative-ordered Kolmogorov-Arnold Networks for Dynamics (KANDy) layers stacked into a machine learning architecture. Compared with SINDy and WSINDy on noisy data, Higher-Order-KANDy learns governing equations without a candidate library (e.g., recovering the full-period $-\sin(u)$ with nothing named in advance), natively for higher-order systems, and holds the correct term structure as noise grows, selecting the exact active set on every seed at noise levels up to 7.5% where an oracle weak-form regression admits a spurious term, though the oracle remains the more accurate coefficient estimator. We leverage the relationship between delay and differential embeddings, along with the Kolmogorov-Arnold representation theorem, to construct a delay-to-jet map and learn it as a Kolmogorov-Arnold network to preserve derivative structure and provide meaningful terms that appear in the governing equation. In our synthetic benchmarks, this construction recovers equations in the presence of coarse sampling and noise.
