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

TL;DR

This paper introduces three innovative multiscale learning approaches using reduced-order models and neural networks to improve the prediction of complex dynamical systems across multiple time scales, especially with limited data.

Contribution

It presents novel methods combining Partition of Unity, SVD, and Sparse High-Order SVD for multiscale modeling and prediction of nonlinear dynamical systems with incomplete observations.

Findings

Effective capture of macro- and micro-scale dynamics

Robust reconstruction from limited measurements

Applicable to high-dimensional, real-world systems

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.