Application of Reduced-Order Models for Temporal Multiscale Representations in the Prediction of Dynamical Systems

TL;DR

This work tackles the prediction of dynamical systems exhibiting slow and fast time scales by introducing three data-driven reduced-order strategies that preserve interpretability while handling incomplete data. The Partition of Unity approach combines macro-scale neural approximations with micro-scale enrichments to learn both global and local dynamics, while SVD-based modal decomposition provides a low-rank separation of macro and micro components, and Sparse High-Order SVD enables multiscale reconstruction from sparse observations via residual-corrected neural factors. Across representative one-dimensional multiscale systems, including Duffing and harmonic oscillator models, the methods demonstrate accurate recovery of both coarse and fine dynamics with competitive efficiency and robustness to data sparsity. Collectively, these approaches offer a scalable, interpretable framework for multiscale dynamical systems and point to extensions with neural ODEs, physics-informed networks, and adaptive partitioning for high-dimensional applications.

Abstract

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.

Figures (13)

• Figure 1: Workflow for Partition of Unity.
• Figure 2: PU approximation for Eq. \ref{['eq:sin&exp']}.
• Figure 3: PU approximation for Eq. \ref{['eq:cos&sin']} using one mode.
• Figure 4: PU approximation for Eq. \ref{['eq:cos&sin']} using two modes.
• Figure 5: PU approximation for Eq. \ref{['eq:duffin']}.