ResKoopNet: Learning Koopman Representations for Complex Dynamics with Spectral Residuals

TL;DR

ResKoopNet addresses the challenge of accurately estimating the Koopman operator spectrum for high-dimensional nonlinear dynamics, including both discrete and continuous components. It directly minimizes the spectral residual by learning dictionary functions with a neural network, yielding a closed-form Koopman matrix $\tilde{K}(\theta)=(G(\theta)+\sigma I)^{-1}A(\theta)$ and enabling pseudospectrum analysis. The method overcomes spectral inclusion and demonstrates superior spectral accuracy across pendulum, turbulence, and neural-dynamics datasets, requiring fewer dictionary observables than prior approaches. Although computationally intensive, this framework integrates spectral accuracy with neural dictionary learning to provide a principled, data-driven tool for analyzing complex dynamical systems.

Abstract

Analyzing the long-term behavior of high-dimensional nonlinear dynamical systems remains a significant challenge. While the Koopman operator framework provides a powerful global linearization tool, current methods for approximating its spectral components often face theoretical limitations and depend on predefined dictionaries. Residual Dynamic Mode Decomposition (ResDMD) advanced the field by introducing the \emph{spectral residual} to assess Koopman operator approximation accuracy; however, its approach of only filtering precomputed spectra prevents the discovery of the operator's complete spectral information, a limitation known as the `spectral inclusion' problem. We introduce ResKoopNet (Residual-based Koopman-learning Network), a novel method that directly addresses this by explicitly minimizing the \emph{spectral residual} to compute Koopman eigenpairs. This enables the identification of a more precise and complete Koopman operator spectrum. Using neural networks, our approach provides theoretical guarantees while maintaining computational adaptability. Experiments on a variety of physical and biological systems show that ResKoopNet achieves more accurate spectral approximations than existing methods, particularly for high-dimensional systems and those with continuous spectra, which demonstrates its effectiveness as a tool for analyzing complex dynamical systems.

Figures (17)

• Figure 1: The optimizer on the right searches for the optimal dictionary by minimizing Eq. \ref{['relative_residual_approximation']}.
• Figure 2: The four plots depict the spectrum of the Koopman operator, constructed using varying dictionary sizes $N_K$ of 25, 50, 100, and 300. Each plot utilizes 90 initial points to illustrate the impact of increasing the dictionary size on approximating the spectrum of the Koopman operator.
• Figure 3: Same example as Figure \ref{['fig:pendulum_90']} but with larger data size, using 240 initial points to show the effect of increasing dictionary size on the Koopman spectrum approximation.
• Figure 4: Comparison with classical methods. Eigenvalue spectra computed from a $300\times300$ Koopman matrix using EDMD, EDMD-DL, Hankel-DMD, and ResDMD.
• Figure 5: Turbulence detection using Koopman modes from 250 observables. (a) shows the original 2D pressure field. (b) shows Koopman mode 1 which has the smallest residual and closely matches the original pressure field. (c) and (d) show the acoustic vibration and turbulent fluctuation by the Koopman modes with the 2nd and 3rd smallest spectral residual. (e) and (f) show the plot of all singular values and first 150 singular values when applying truncated SVD method, respectively.

Theorems & Definitions (6)

• Remark 3.1
• Remark 3.2
• Remark A.1
• Theorem A.2: Direct Approximation Theorem, $L^2$-version
• Remark A.3
• Remark A.5