arXiv 2606.199842026-07-03cs.LG

Kolmogorov-Arnold 储层计算

Kolmogorov-Arnold Reservoir Computing

Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China（电子科技大学基础与前沿科学研究所）；Potsdam Institute for Climate Impact Research（波茨坦气候影响研究所）；Department of Physics, Humboldt University Berlin（柏林洪堡大学物理系）；Research Institute of Intelligent Complex Systems, Fudan University（复旦大学智能复杂系统研究所）；School of Physics, University of Electronic Science and Technology of China（电子科技大学物理学院）；Key Laboratory of Quantum Physics and Photonic Quantum Information, Ministry of Education, University of Electronic Science and Technology of China（电子科技大学教育部量子物理与光子量子信息重点实验室）；Non-classical Information Science Basic Discipline Research Center of Sichuan Province, University of Electronic Science and Technology of China（电子科技大学四川省非经典信息科学基础学科研究中心）

Juntian Huang, Jürgen Kurths, Ying Tang

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AI总结：

提出Kolmogorov-Arnold储层计算（KARC），用显式基函数展开替代储层，结合KAN的表达能力和储层计算的闭式训练，在偏微分方程等基准上优于现有方法。

摘要：

储层计算为预测动力系统提供了轻量级框架，但由于表示能力有限，可能难以捕捉长程依赖。传统储层计算循环使用可训练储层，对超参数敏感，而下一代储层计算以特征维度快速增长为代价去除了循环。在此，我们开发了Kolmogorov-Arnold储层计算（KARC），它受Kolmogorov-Arnold表示定理启发，用显式基函数展开替代储层。我们严格证明KARC是Kolmogorov-Arnold网络（KAN）的轻量级设计，保留了KAN的潜在表达能力，同时允许储层计算的高效闭式训练。在相当的成本下，KARC在包括偏微分方程在内的挑战性基准上优于现有储层计算方法。它还可以与生成扩散模型集成用于文本到图像生成。因此，本工作建立了储层计算与KAN之间的原则性桥梁，实现了高效高保真的动力系统预测。

英文摘要：

Reservoir computing offers a lightweight framework for forecasting dynamical systems but may struggle to capture long-range dependencies due to limited representational capacity. Conventional reservoir computing recurrently uses fixed reservoirs with hyperparameter sensitivity, while the next generation reservoir computing removes recurrence at the cost of rapidly growing feature dimensions. Here, we develop Kolmogorov-Arnold Reservoir Computing (KARC), which replaces reservoirs with explicit basis-function expansions inspired by the Kolmogorov-Arnold representation theorem. We rigorously show that KARC is a lightweight design of Kolmogorov-Arnold networks (KANs), preserving the potential expressive capacity of KANs while admitting efficient closed-form training of reservoir computing. At comparable cost, KARC outperforms existing reservoir computing methods on challenging benchmarks including partial differential equations. It can also be integrated with generative diffusion models for facilitating text-to-image generation. This work thus establishes a principled bridge between reservoir computing and KANs, yielding a unified framework for efficient dynamical forecasting and generative modeling.

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