Data-driven model discovery of complex dynamical systems is typically done using sparse optimization, but it has a fundamental limitation: sparsity in that the underlying governing equations of the system contain only a small number of elementary mathematical terms. Examples where sparse optimization fails abound, such as the classic Ikeda or optical-cavity map in nonlinear dynamics and a large variety of ecosystems. Exploiting the recently articulated Kolmogorov-Arnold networks, we develop a general model-discovery framework for any dynamical systems including those that do not satisfy the sparsity condition. In particular, we demonstrate non-uniqueness in that a large number of approximate models of the system can be found which generate the same invariant set with the correct statistics such as the Lyapunov exponents and Kullback-Leibler divergence. An analogy to shadowing of numerical trajectories in chaotic systems is pointed out.

本研究解决了复杂动力系统模型发现中的稀疏优化局限性，提出了一种基于Kolmogorov-Arnold网络的通用模型发现框架，适用于不满足稀疏条件的动力系统。研究指出存在多个近似模型可生成相同的不变集及准确统计特征，从而影响系统建模的理解和应用。

基于数据驱动模型发现的Kolmogorov-Arnold网络