KAN-ODEs：用于学习动力系统与隐藏物理的Kolmogorov-Arnold网络常微分方程
KAN-ODEs: Kolmogorov-Arnold Network Ordinary Differential Equations for Learning Dynamical Systems and Hidden Physics
AI总结：
本文提出KAN-ODEs，将Kolmogorov-Arnold网络作为神经ODE主干，用于学习动力系统和隐藏物理，相比传统神经ODE在精度、可解释性和参数效率上更优，并在多个复杂方程中验证了其有效性。
摘要：
Kolmogorov-Arnold网络（KANs）作为多层感知器（MLPs）的替代方案，是近期的一项发展，展示了在数据驱动建模方面的强大潜力。本工作将KANs作为神经常微分方程（ODE）框架的主干，将其应用推广到动力系统和科学机器学习应用中常见的时间依赖及时间网格敏感的情况。所提出的KAN-ODEs保留了神经ODE的灵活动力系统建模框架，同时利用了KANs相对于MLPs的诸多优势，包括更高的精度和更快的神经缩放、更强的可解释性和泛化能力，以及更低的参数数量。首先，我们在经典Lotka-Volterra捕食者-猎物模型的综合研究中定量证明了这些改进。随后，我们展示了KAN-ODE框架在更高复杂度和数据稀疏场景中学习符号源项和完整解剖面的能力，包括波传播和激波形成、复薛定谔方程以及Allen-Cahn相分离方程。KAN-ODEs的成功训练，以及与传统神经ODE相比的性能提升，意味着这一新颖网络架构在众多科学机器学习应用中具有巨大潜力，可用于发现隐藏物理和预测动态演化。
英文摘要：
Kolmogorov-Arnold networks (KANs) as an alternative to multi-layer perceptrons (MLPs) are a recent development demonstrating strong potential for data-driven modeling. This work applies KANs as the backbone of a neural ordinary differential equation (ODE) framework, generalizing their use to the time-dependent and temporal grid-sensitive cases often seen in dynamical systems and scientific machine learning applications. The proposed KAN-ODEs retain the flexible dynamical system modeling framework of Neural ODEs while leveraging the many benefits of KANs compared to MLPs, including higher accuracy and faster neural scaling, stronger interpretability and generalizability, and lower parameter counts. First, we quantitatively demonstrated these improvements in a comprehensive study of the classical Lotka-Volterra predator-prey model. We then showcased the KAN-ODE framework's ability to learn symbolic source terms and complete solution profiles in higher-complexity and data-lean scenarios including wave propagation and shock formation, the complex Schrödinger equation, and the Allen-Cahn phase separation equation. The successful training of KAN-ODEs, and their improved performance compared to traditional Neural ODEs, implies significant potential in leveraging this novel network architecture in myriad scientific machine learning applications for discovering hidden physics and predicting dynamic evolution.
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。
