fSRD：模糊谱区域分解——通过自适应谱学习架构实现自动多算子库普曼表示
fSRD: Fuzzy Spectral Region Decomposition -- Automated Multi Operator Koopman Representations via an Adaptive Spectral Learning Architecture
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中文总结 AI 辅助
针对高度非线性混沌动力系统建模难题，提出模糊谱区域分解（fSRD）方法，通过多个算子估计有限库普曼表示，实现数据自适应框架组装局部不变嵌入，在多系统和数据条件下展现出强预测性、可解释性和表现力。
中文摘要 AI 辅助
高度非线性混沌动力系统因复杂度、表现力和数据效率之间的基本权衡而难以建模。现代机器学习方法虽有强大预测性能，但常依赖先验系统知识或可解释性有限的精选数据。库普曼算子理论提供了一个有前景的方向，但许多数据驱动的库普曼方法难以识别有用的有限维谱嵌入。为克服这些限制，我们引入模糊谱区域分解（fSRD），这是一个通过多个算子估计有限库普曼表示的全自动学习框架。该方法实现了一个数据自适应框架来组装局部不变嵌入，即不变分解。fSRD在学习非线性系统诱导演化算子的有限维表示时，能实现高精度的线性重构，将可解释的算子理论模型与富有表现力的数据驱动序列学习相联系。这些嵌入通过全局模糊树模型自适应构建，借鉴模糊神经架构学习诱导动力学并优先考虑简约解。在典型混沌系统（如洛伦兹和杜芬系统）及高维真实世界数据上的实证结果表明，该方法在数据丰富和数据有限的情况下都具有强大的预测准确性、可解释性和稳健表现力，凸显了其通用性。
英文摘要
Highly nonlinear chaotic dynamical systems remain difficult to model due to fundamental trade-offs between complexity, expressivity, and data efficiency. Modern machine learning methods achieve strong predictive performance but often rely on a-priori system knowledge or curated data with limited interpretability. Koopman operator theory offers a promising direction via linear representation in an infinite-dimensional observable space. However, many data-driven Koopman methods seek globally valid operators for which useful finite-dimensional spectral embeddings remain difficult to identify under these constraints. To overcome associated limitations, we introduce Fuzzy Spectral Region Decomposition (fSRD), a fully automated learning framework for estimating finite Koopman representation via multiple operators. The proposed method realizes a data-adaptive framework for assembling locally invariant embeddings, termed Invariant Decomposition. fSRD achieves highly accurate linear reconstructions of nonlinear systems while learning finite-dimensional representations of their induced evolution operators, bridging interpretable operator-theoretic models with expressive data-driven sequence learning. These embeddings are adaptively constructed via a global fuzzy tree model, drawing inspiration from fuzzy neural architectures to learn the induced dynamics while prioritizing parsimonious solutions. Empirical results across canonical chaotic systems (e.g., Lorenz and Duffing) and high-dimensional real-world data demonstrate strong predictive accuracy, interpretability, and robust expressivity across data-rich and data-limited regimes, highlighting the method's generality.
