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

2025-10-21

AI Summary

Addressing challenges in modeling complex multiscale dynamical systems—including strong nonlinearity, sensitivity to initial conditions, difficulty capturing high-frequency dynamics, and incomplete observations—this paper proposes an interpretable multiscale learning framework. Methodologically, it integrates partition of unity (PU), singular value decomposition (SVD), and sparse higher-order SVD, coupled with neural networks: PU enables local macro-micro协同 prediction; SVD isolates dominant modes to decouple multiscale behaviors; sparse higher-order SVD robustly reconstructs high-dimensional dynamics from limited or incomplete observations. The framework supports cross-scale modeling and full-scale dynamic approximation, achieving high accuracy while ensuring strong scalability and physical interpretability. Empirical results demonstrate significantly improved prediction reliability and generalization capability in practical scenarios involving sparse sensing and partial observability.

Technology Category

• Machine Learning: Multimodal Learning
• Intelligent Robots: Multimodal Perception & Sensor Fusion
• Computer Vision: Multi-modal Vision

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile 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.

Problem

• Predicting complex multiscale dynamical systems with nonlinearities
• Overcoming limitations in capturing high-frequency behaviors
• Reconstructing multiscale dynamics from limited measurements

Innovation

• Partition of Unity with neural networks for multiscale dynamics
• Singular Value Decomposition to separate macro and micro scales
• Sparse High-Order SVD for limited measurements reconstruction