ResKoopNet: Learning Koopman Representations for Complex Dynamics with Spectral Residuals

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.

Understanding how complex systems change over time is a major scientific challenge. Current methods try to find underlying simple patterns in these systems but often get misled by missing crucial information. We've developed a new data-driven approach called ResKoopNet that acts like an intelligent pattern-finder. It automatically learns the best way to view a complex system to reveal its hidden properties and constantly works to minimize any errors in the patterns it detects. We tested ResKoopNet on various complex systems, from physical models to biological brain activities, and found it significantly more accurate at uncovering the true, complete patterns. This could lead to better ways to analyze and predict the behavior of many complex systems in science and engineering.