用降阶模型分解复杂系统多尺度动态，提升预测精度。
Application of Reduced-Order Models for Temporal Multiscale Representations in the Prediction of Dynamical Systems
- 结合神经网络与单位分解法，分块预测宏观与微观行为。
- 通过奇异值分解提取主导模式，明确分离高低频动态。
- 稀疏高阶SVD可从有限观测重建多尺度演化，适合不完整数据。
复杂多尺度系统的建模与预测因非线性特性、对初值敏感以及传统机器学习方法难以捕捉高频行为而面临挑战。为此，本文提出三种多尺度学习方法：其一，利用单位分解（PU）方法结合神经网络，将动态分解为局部成分，直接预测宏观与微观行为；其二，采用奇异值分解（SVD）提取主导模态，显式分离宏观与微观动态；由于实际中难以获取完整数据矩阵，第三种方法引入稀疏高阶SVD，从有限测量中重构多尺度动态。上述方法共同确保粗粒度与细粒度动态均被准确捕捉，使该框架在涉及复杂多尺度现象的真实场景中具备有效性，并能适应高维系统及不完备观测，实现对研究现象中所有时间尺度的近似与解释。
原文摘要 · Abstract (English)
Modeling and predicting the dynamics of complex multiscale systems remains a significant challenge due to their inherent nonlinearities and sensitivity to initial conditions, as well as limitations of traditional machine learning methods that fail to capture high frequency behaviours. To overcome these difficulties, we propose three approaches for multiscale learning. The first leverages the Partition of Unity (PU) method, integrated with neural networks, to decompose the dynamics into local components and directly predict both macro- and micro-scale behaviors. The second applies the Singular Value Decomposition (SVD) to extract dominant modes that explicitly separate macro- and micro-scale dynamics. Since full access to the data matrix is rarely available in practice, we further employ a Sparse High-Order SVD to reconstruct multiscale dynamics from limited measurements. Together, these approaches ensure that both coarse and fine dynamics are accurately captured, making the framework effective for real-world applications involving complex, multi-scale phenomena and adaptable to higher-dimensional systems with incomplete observations, by providing an approximation and interpretation in all time scales present in the phenomena under study.
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