ResKoopNet: Learning Koopman Representations for Complex Dynamics with Spectral Residuals

• URL: http://arxiv.org/abs/2501.00701v3
• Date: Tue, 01 Apr 2025 19:41:00 GMT
• Title: ResKoopNet: Learning Koopman Representations for Complex Dynamics with Spectral Residuals
• Authors: Yuanchao Xu, Kaidi Shao, Nikos Logothetis, Zhongwei Shen,
• Abstract summary: Existing methods for approximating Koopman operator's spectral components often suffer from theoretical limitations.<n>We introduce ResKoopNet, a novel method that explicitly minimizes the spectral residual to compute Koopman eigenpairs.<n>Experiments on physical and biological systems demonstrate ResKoopNet's superior accuracy in spectral approximation compared to existing methods.
• Score: 1.8570740863168362
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Analyzing long-term behaviors in high-dimensional nonlinear dynamical systems remains challenging, with the Koopman operator framework providing a powerful global linearization approach, though existing methods for approximating its spectral components often suffer from theoretical limitations and reliance on predefined dictionaries. While Residual Dynamic Mode Decomposition (ResDMD) introduced the spectral residual to assess the accuracy of Koopman operator approximation, its only filters precomputed spectra, which prevents it from fully discovering the Koopman operator's complete spectral information (a limitation sometimes referred to as the 'spectral inclusion' problem). We introduce ResKoopNet (Residual-based Koopman-learning Network), a novel method that addresses this limitation by explicitly minimizing the spectral residual to compute Koopman eigenpairs, which can identify a more precise and complete spectrum of the Koopman operator. This approach provides theoretical guarantees while maintaining computational adaptability through a neural network implementation. Experiments on physical and biological systems demonstrate ResKoopNet's superior accuracy in spectral approximation compared to existing methods, particularly for systems with continuous spectra and high dimensional, which makes it as an effective tool for analyzing complex dynamical systems.

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