KoopGen: Koopman Generator Networks for Representing and Predicting Dynamical Systems with Continuous Spectra
This addresses a fundamental problem in dynamical systems and machine learning for researchers and practitioners, offering a novel method for handling continuous spectra, though it builds on existing Koopman-based approaches.
The paper tackles the challenge of representing and predicting high-dimensional chaotic dynamical systems with continuous spectra by introducing KoopGen, a neural Koopman framework that models dynamics through structured Koopman generators, improving prediction accuracy and stability across systems like nonlinear oscillators and spatiotemporal dynamics.
Representing and predicting high-dimensional and spatiotemporally chaotic dynamical systems remains a fundamental challenge in dynamical systems and machine learning. Although data-driven models can achieve accurate short-term forecasts, they often lack stability, interpretability, and scalability in regimes dominated by broadband or continuous spectra. Koopman-based approaches provide a principled linear perspective on nonlinear dynamics, but existing methods rely on restrictive finite-dimensional assumptions or explicit spectral parameterizations that degrade in high-dimensional settings. Against these issues, we introduce KoopGen, a generator-based neural Koopman framework that models dynamics through a structured, state-dependent representation of Koopman generators. By exploiting the intrinsic Cartesian decomposition into skew-adjoint and self-adjoint components, KoopGen separates conservative transport from irreversible dissipation while enforcing exact operator-theoretic constraints during learning. Across systems ranging from nonlinear oscillators to high-dimensional chaotic and spatiotemporal dynamics, KoopGen improves prediction accuracy and stability, while clarifying which components of continuous-spectrum dynamics admit interpretable and learnable representations.