用新型神经网络发现复杂系统规律，突破传统方法对简单公式的依赖。
Data-driven model discovery with Kolmogorov-Arnold networks
- 基于柯尔莫哥洛夫-阿诺德网络构建通用建模框架
- 能准确复现混沌系统的统计特征如李雅普诺夫指数
- 适合研究非稀疏、复杂动力学系统，解释性优于普通神经网络
数据驱动的动力系统建模通常依赖稀疏优化，但其核心假设——系统方程仅由少量基本数学项构成——在许多实际系统中不成立，如非线性动力学中的经典Ikeda映射和大量生态系统。本文利用近期提出的柯尔莫哥洛夫-阿诺德网络（KAN），构建了一个适用于任意动力系统的通用模型发现框架，特别针对不满足稀疏性条件的系统。我们证明了模型的非唯一性：存在大量近似模型可生成相同的不变集，并具备一致的统计特性，如李雅普诺夫指数和相对熵（Kullback-Leibler divergence）。这一现象类比于混沌系统中数值轨迹的影子性质。
原文摘要 · Abstract (English)
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.
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