Application of Reduced-Order Models for Temporal Multiscale Representations in the Prediction of Dynamical Systems
This addresses the problem of modeling nonlinear multiscale systems for researchers in computational science and engineering, though it appears incremental as it builds on existing methods like Partition of Unity and SVD.
The authors tackled the challenge of predicting complex multiscale dynamical systems by proposing three reduced-order model approaches that decompose dynamics into macro- and micro-scale components, resulting in accurate capture of both coarse and fine behaviors for real-world applications.
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.