用储层计算思想解决非线性系统线性化难题，提升稳定性与精度。
Koopman Identification of Nonlinear Systems via Reservoir Liftings
- 将储层视为可调控时序记忆的有限维库，通过谱半径控制记忆深度。
- 在合成数据上重建精度优于EDMD和基于汉克尔的提升方法，且动态更稳定。
- 适合研究非线性动力系统建模、控制与预测的科研人员使用。
通过柯普曼算子理论学习非线性动力系统的可处理线性表示，常受字典选择、时间记忆编码和数值病态性阻碍。受储层计算（RC）范式启发，本文提出RC-Koopman框架，将储层解释为具有状态依赖、有限维的柯普曼字典，其时间深度由谱半径显式控制。我们证明了回声状态性质（ESP）保证了提升后柯普曼近似的适定性和良好数值条件。基于相关性的谱半径选择算法使储层记忆与系统主导时间尺度对齐。分析揭示了储层有限记忆如何决定哪些柯普曼特征函数能从提升特征中可观测。在合成基准测试上的评估表明，相比扩展动态模态分解（EDMD）和基于汉克尔的提升方法，RC-Koopman在底层非线性动力学重建精度与动态稳定性之间取得更优平衡。代码已公开：https://github.com/NEAR-the-future/RC-Koopman.git
原文摘要 · Abstract (English)
Learning tractable linear representations of nonlinear dynamical systems via Koopman operator theory is often hindered by dictionary selection, temporal memory encoding, and numerical ill-conditioning. Inspired by Reservoir Computing (RC) paradigm, this paper introduces the RC-Koopman framework, which interprets reservoir as a stateful, finite-dimensional Koopman dictionary whose temporal depth is explicitly controlled by its spectral radius. We show that the Echo State Property (ESP) guarantees well-posedness and favorable numerical conditioning of the lifted Koopman approximation. A correlation-based spectral radius selection algorithm aligns reservoir memory with dominant system timescales. Analysis reveals how the finite memory of the reservoir determines which Koopman eigenfunctions remain observable from the lifted features. Evaluation on synthetic benchmarks demonstrates that RC-Koopman achieves a favorable balance between reconstruction accuracy of the underlying nonlinear dynamics and dynamical stability, compared to Extended Dynamic Mode Decomposition (EDMD) and Hankel-based lifting approaches. Code available at: https://github.com/NEAR-the-future/RC-Koopman.git
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