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.

翻译：表示和预测高维时空混沌动力系统仍然是动力系统和机器学习领域的一个基本挑战。尽管数据驱动模型可以实现精确的短期预测，但在宽带或连续谱主导的体系中，它们往往缺乏稳定性、可解释性和可扩展性。基于Koopman的方法为非线性动力学提供了一个原理性的线性视角，但现有方法依赖于有限维的严格假设或显式的谱参数化，这些在高维设置下性能会下降。针对这些问题，我们提出了KoopGen，这是一个基于生成器的神经Koopman框架，它通过结构化的、状态依赖的Koopman生成器表示来建模动力学。通过利用内在的笛卡尔分解（分解为斜自伴和自伴分量），KoopGen将保守输运与不可逆耗散分离开来，同时在训练过程中强制执行精确的算子理论约束。在从非线性振荡器到高维混沌和时空动力学的多种系统中，KoopGen提高了预测精度和稳定性，同时阐明了连续谱动力学的哪些部分允许可解释和可学习的表示。