用生成模型提升复杂系统预测的稳定性与可解释性
KoopGen: Koopman Generator Networks for Representing and Predicting Dynamical Systems with Continuous Spectra
- 基于神经网络构建可学习的Koopman生成器，分离保守与耗散动态
- 在高维混沌系统上实现更准确、更稳定的长期预测
- 适合研究复杂动力系统建模的科研人员
高维且时空混沌的动力系统表征与预测仍是动力系统与机器学习中的基础挑战。尽管数据驱动模型可实现高精度短期预测，但在宽带或连续谱主导的场景下常缺乏稳定性、可解释性与可扩展性。基于Koopman的方法为非线性动力学提供了严谨的线性视角，但现有方法依赖于有限维假设或显式谱参数化，在高维场景中性能下降。针对这些问题，我们提出KoopGen——一种基于生成器的神经Koopman框架，通过状态相关的结构化表示建模Koopman生成器。利用其内在的斜自伴与自伴分量分解，KoopGen分离保守输运与不可逆耗散，并在学习过程中强制满足精确的算子理论约束。在从非线性振子到高维混沌与时空动力系统的多类系统上，KoopGen显著提升了预测精度与稳定性，同时揭示了哪些连续谱动态成分具有可解释且可学习的表征能力。
原文摘要 · Abstract (English)
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.
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