NN-ResDMD: Learning Koopman Representations for Complex Dynamics with Spectral Residuals

2025-01-01

AI Summary

To address the insufficient reliability of Koopman operator learning for long-term behavior analysis of high-dimensional nonlinear dynamical systems, this paper proposes NN-ResDMD—a novel end-to-end neural network framework. Departing from conventional ResDMD, which relies on handcrafted basis functions and pre-specified spectral structures, NN-ResDMD introduces a spectrum-residual-driven unsupervised learning paradigm: it directly minimizes the Koopman spectral residual, jointly optimizing the operator, eigenfunctions, and spectral structure. Theoretical analysis guarantees convergence and interpretability, while empirical evaluation demonstrates strong generalization. Experiments across diverse physical and biological systems show significant improvements in spectral estimation accuracy and computational scalability, more robust convergence, and over 40% higher false-mode rejection rate compared to baselines.

Technology Category

• Machine Learning: Learning with Manifolds
• Intelligent Robots: State Estimation
• Search and Optimization: Learning to Search

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• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Analyzing long-term behaviors in high-dimensional nonlinear dynamical systems remains a significant challenge. The Koopman operator framework has emerged as a powerful tool to address this issue by providing a globally linear perspective on nonlinear dynamics. However, existing methods for approximating the Koopman operator and its spectral components, particularly in large-scale systems, often lack robust theoretical guarantees. Residual Dynamic Mode Decomposition (ResDMD) introduces a spectral residual measure to assess the convergence of the estimated Koopman spectrum, which helps filter out spurious spectral components. Nevertheless, it depends on pre-computed spectra, thereby inheriting their inaccuracies. To overcome its limitations, we introduce the Neural Network-ResDMD (NN-ResDMD), a method that directly estimates Koopman spectral components by minimizing the spectral residual. By leveraging neural networks, NN-ResDMD automatically identifies the optimal basis functions of the Koopman invariant subspace, eliminating the need for manual selection and improving the reliability of the analysis. Experiments on physical and biological systems demonstrate that NN-ResDMD significantly improves both accuracy and scalability, making it an effective tool for analyzing complex dynamical systems.

Problem

• Reliability
• Koopman Operator
• ResDMD Method

Innovation

• NN-ResDMD
• Koopman Operator
• Automated Optimization